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Search Results (840)

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9 pages, 1346 KiB  
Proceeding Paper
From Grinding to Green Energy: Pursuit of Net-Zero Emissions in Cement Production
by Md. Shahariar Ahmed, Anica Tasnim and Golam Kabir
Eng. Proc. 2024, 76(1), 8; https://doi.org/10.3390/engproc2024076008 - 15 Oct 2024
Abstract
In an age of heightened environmental awareness and the pressing need for net-zero emissions, concerns over rising energy consumption in cement production, responsible for 5–8% of global CO2 emissions, have intensified. This paper proposes a novel pioneering framework that integrates Shannon’s entropy [...] Read more.
In an age of heightened environmental awareness and the pressing need for net-zero emissions, concerns over rising energy consumption in cement production, responsible for 5–8% of global CO2 emissions, have intensified. This paper proposes a novel pioneering framework that integrates Shannon’s entropy and Multi-Criteria Decision Making (MCDM) methods to steer the cement industry towards sustainability and net-zero emissions. Utilizing Shannon’s entropy, the research impartially determines the significance of multiple criteria, reducing biases in decision-making for energy efficiency in cement production. Four MCDM methods (TOPSIS, VIKOR, ELECTRE, WSM) are applied to rank energy efficiency alternatives, providing a nuanced analysis of options for the cement industry. The study integrates sensitivity analysis to evaluate the robustness of MCDM methods under varying conditions, assessing the impact of changes in criteria weights on the ranking of energy efficiency alternatives and showcasing the adaptability of the proposed framework. Examining six diverse scenarios reveals the framework’s adaptability and the versatility of the Horizontal Roller Mill (HRM), with the Vertical Roller Mill (VRM) emerging as a cost-effective emission reduction alternative. This interdisciplinary approach, integrating information theory, decision science, and environmental engineering, extends beyond industry relevance, providing valuable insights aligned with global sustainability goals. Harmonizing economic viability with ecological responsibility, this report offers an instructive guide, propelling the cement industry toward a more sustainable future. Full article
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<p>Flow diagram of energy-intensive cement [<a href="#B2-engproc-76-00008" class="html-bibr">2</a>,<a href="#B4-engproc-76-00008" class="html-bibr">4</a>] production (wet process).</p>
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<p>Electricity consumption in various cement-making processes [<a href="#B3-engproc-76-00008" class="html-bibr">3</a>]. Distribution of energy among equipment used in cement manufacturing [<a href="#B4-engproc-76-00008" class="html-bibr">4</a>].</p>
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<p>Study framework.</p>
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<p>Ranking comparison with the method of paper.</p>
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10 pages, 1209 KiB  
Article
Simple Sequence Repeat Marker-Based Genetic Diversity and Chemical Composition Analysis of Ancient Camellia sinensis in Jiulong County, Sichuan Province, China
by Haitao Huang, Shuwen He, Xuxia Zheng, Daliang Shi, Peixian Bai, Yun Zhao, Jizhong Yu and Xiaojun Niu
Genes 2024, 15(10), 1317; https://doi.org/10.3390/genes15101317 - 14 Oct 2024
Viewed by 292
Abstract
Background/Objectives: The ancient tea plant germplasm resources are rich in genetic diversity and provide an important basis for the genetic diversity in tea germplasm resources. To explore the genetic diversity of ancient tea plant germplasm resources in Jiulong County, Sichuan Province. Methods: 59 [...] Read more.
Background/Objectives: The ancient tea plant germplasm resources are rich in genetic diversity and provide an important basis for the genetic diversity in tea germplasm resources. To explore the genetic diversity of ancient tea plant germplasm resources in Jiulong County, Sichuan Province. Methods: 59 ancient tea tree germplasm resources were analyzed using simple sequence repeat (SSR) molecular markers and chemical composition analysis. Results: The results showed that a total of 83 alleles were amplified by 23 pairs of SSR primers, with an average observed allele number (Na) of 3.6 and an effective allele number (Ne) of 2.335. The average Shannon information index (I) and the polymorphic information content (PIC) of the primers were 0.896 and 0.446, respectively. The results of the UPGMA cluster analysis showed that 59 ancient tea tree samples could be classified into five different subgroups. Based on the results of chemical composition analysis, two specific tea germplasm resources with high amino acid content, 10 excellent germplasm resources with tea polyphenol content over 20% and some other tea germplasm resources were identified. Conclusions: This study reveals that Jiulong’s ancient tea tree germplasm exhibits significant genetic diversity and includes valuable tea tree planting resources. These findings provide a foundational framework for the conservation, detailed exploration and sustainable utilization of these resources. Full article
(This article belongs to the Special Issue Genetic and Genomic Studies of Crop Breeding)
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<p>Genetic structure analysis of 59 ancient tea tree samples in Jiulong County. (<b>a</b>) Relationship between K and ΔK. (<b>b</b>) Model based on population structure analysis of 59 evaluated germplasms (K = 3).</p>
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<p>UPGMA evolutionary tree analysis of 59 samples.</p>
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<p>Frequency of distributions of amino acid and catechins in 59 ancient tea resources of Jiulong.</p>
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15 pages, 1956 KiB  
Article
Information–Theoretic Analysis of Visibility Graph Properties of Extremes in Time Series Generated by a Nonlinear Langevin Equation
by Luciano Telesca and Zbigniew Czechowski
Mathematics 2024, 12(20), 3197; https://doi.org/10.3390/math12203197 - 12 Oct 2024
Viewed by 222
Abstract
In this study, we examined how the nonlinearity α of the Langevin equation influences the behavior of extremes in a generated time series. The extremes, defined according to run theory, result in two types of series, run lengths and surplus magnitudes, whose complex [...] Read more.
In this study, we examined how the nonlinearity α of the Langevin equation influences the behavior of extremes in a generated time series. The extremes, defined according to run theory, result in two types of series, run lengths and surplus magnitudes, whose complex structure was investigated using the visibility graph (VG) method. For both types of series, the information measures of the Shannon entropy measure and Fisher Information Measure were utilized for illustrating the influence of the nonlinearity α on the distribution of the degree, which is the main topological parameter describing the graph constructed by the VG method. The main finding of our study was that the Shannon entropy of the degree of the run lengths and the surplus magnitudes of the extremes is mostly influenced by the nonlinearity, which decreases with with an increase in α. This result suggests that the run lengths and surplus magnitudes of extremes are characterized by a sort of order that increases with increases in nonlinearity. Full article
(This article belongs to the Special Issue Recent Advances in Time Series Analysis)
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<p>Graphs of the potentials <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mi>α</mi> </semantics></math> = 1.0, 2.0, and 3.0.</p>
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<p>Definition of extremes: after fixing the threshold <span class="html-italic">T</span> (blue horizontal line), a run is the sequence of <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>L</mi> </mrow> </semantics></math> contiguous values (red-filled circles) above the threshold. The surplus magnitude <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>M</mi> </mrow> </semantics></math> is the sum of the values within the run minus the threshold. In the shown sketch, there are two extremes starting at <span class="html-italic">t</span> = 13 and <span class="html-italic">t</span> = 77, respectively, with the run length <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>L</mi> </mrow> </semantics></math> of 11 and 12 and the surplus magnitude <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>M</mi> </mrow> </semantics></math> of 1.88 and 2.36.</p>
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<p>Sketch of the VG method applied to the first 50 values of the RL (<b>a</b>) and SM (<b>b</b>) of extremes from a time series generated with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>92.5</mn> <mo>%</mo> </mrow> </semantics></math>. The segments represent the connections among the values.</p>
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<p>Relationship between the surplus magnitude and run length. Different colors represent different realizations.</p>
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<p>The average of the slope of the linear fit between Sm and RL for different values of <math display="inline"><semantics> <mi>α</mi> </semantics></math> and different thresholds.</p>
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<p>Mean surplus magnitude versus the mean run length for different values of <math display="inline"><semantics> <mi>α</mi> </semantics></math> and the 92.5% (blue), 95.0% (red), and 97.5% (green) thresholds.</p>
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<p>Case threshold = 92.5%. Left panel (from top to bottom): the density of degree of run length sequences <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>R</mi> <mi>L</mi> </mrow> </msub> </semantics></math>; the Fisher–Shannon Information Plane of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>R</mi> <mi>L</mi> </mrow> </msub> </semantics></math>; and the &lt;FIM&gt; versus &lt;SH&gt; of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>R</mi> <mi>L</mi> </mrow> </msub> </semantics></math>. Right panel (from top to bottom): the density of degree of surplus magnitude sequences <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>S</mi> <mi>M</mi> </mrow> </msub> </semantics></math>; the Fisher–Shannon Information Plane of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>S</mi> <mi>M</mi> </mrow> </msub> </semantics></math>; and the &lt;FIM&gt; versus &lt;SH&gt; of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>S</mi> <mi>M</mi> </mrow> </msub> </semantics></math>.</p>
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<p>Case threshold = 95.0%. Left panel (from top to bottom): the density of degree of the run length sequences <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>R</mi> <mi>L</mi> </mrow> </msub> </semantics></math>; the Fisher–Shannon Information Plane of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>R</mi> <mi>L</mi> </mrow> </msub> </semantics></math>; and the &lt;FIM&gt; versus &lt;SH&gt; of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>R</mi> <mi>L</mi> </mrow> </msub> </semantics></math>. Right panel (from top to bottom): the density of degree of the surplus magnitude sequences <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>S</mi> <mi>M</mi> </mrow> </msub> </semantics></math>; the Fisher–Shannon Information Plane of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>S</mi> <mi>M</mi> </mrow> </msub> </semantics></math>; and the &lt;FIM&gt; versus &lt;SH&gt; of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>S</mi> <mi>M</mi> </mrow> </msub> </semantics></math>.</p>
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<p>Case threshold = 97.5%. Left panel (from top to bottom): the density of degree of the run length sequences <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>R</mi> <mi>L</mi> </mrow> </msub> </semantics></math>; the Fisher–Shannon Information Plane of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>R</mi> <mi>L</mi> </mrow> </msub> </semantics></math>; the &lt;FIM&gt; versus &lt;SH&gt; of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>R</mi> <mi>L</mi> </mrow> </msub> </semantics></math>. Right panel (from top to bottom): the density of degree of the surplus magnitude sequences <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>S</mi> <mi>M</mi> </mrow> </msub> </semantics></math>; the Fisher–Shannon Information Plane of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>S</mi> <mi>M</mi> </mrow> </msub> </semantics></math>; and the &lt;FIM&gt; versus &lt;SH&gt; of <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>S</mi> <mi>M</mi> </mrow> </msub> </semantics></math>.</p>
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12 pages, 3022 KiB  
Article
Genetic Diversity and Population Structure of Endangered Orchid Cypripedium flavum in Fragmented Habitat Using Fluorescent AFLP Markers
by Shijun Hu, Meizhen Wang, Xiaohui Yan and Xiaomao Cheng
Plants 2024, 13(20), 2851; https://doi.org/10.3390/plants13202851 (registering DOI) - 11 Oct 2024
Viewed by 316
Abstract
Genetic diversity is crucial for determining the evolutionary potential of a species and is essential for developing optimal conservation strategies. The impact of habitat fragmentation on the genetic diversity of food-deceptive orchids seems to be unpredictable because of their specialized seed and pollen [...] Read more.
Genetic diversity is crucial for determining the evolutionary potential of a species and is essential for developing optimal conservation strategies. The impact of habitat fragmentation on the genetic diversity of food-deceptive orchids seems to be unpredictable because of their specialized seed and pollen dispersal mechanisms. The habitat of deceptive Cypripedium flavum was severely fragmented during the past half century. This study investigated the genetic diversity and structure of seven fragmented Cypripedium flavum populations in Shangrila County using AFLP markers. A total of 376 alleles were identified, with a range of 70 to 81 alleles per locus. The species exhibited considerable genetic diversity, as evidenced by an average Nei’s gene diversity (H) of 0.339 and a Shannon’s information index (I) of 0.505, with all loci being polymorphic. Based on Molecular Variance (AMOVA), 8.75% of the genetic differentiation was found among populations, while the remaining 91.25% of genetic variation occurred within populations. Population structure analysis revealed that the C. flavum germplasm can be categorized into 2 distinct groups, among which there was significant gene flow. Despite habitat fragmentation, C. flavum still retained a high level of genetic diversity, and the substantial gene flow (5.0826) is a key factor in maintaining the genetic diversity. These findings offer valuable insights for the conservation and potential use of C. flavum genetic resources. Full article
(This article belongs to the Special Issue Genetic Diversity and Population Structure of Plants)
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<p>Determinations of genetic groups (K) of <span class="html-italic">C. flavum</span> population.</p>
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<p>Q value distribution of <span class="html-italic">C. flavum</span> K = 2. The 150 individuals were divided into 2 genetic groups. Different color means different genetic groups.</p>
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<p>Principal coordinate analysis using AFLP markers and separation on a two-dimensional diagram. (<b>A</b>) PCA at the population level; (<b>B</b>) PCA at the individual level.</p>
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<p>Map of sampling populations and polymorphic flowers of <span class="html-italic">C. flavum</span>.</p>
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14 pages, 3147 KiB  
Article
Sex-Dependent Rhizosphere Microbial Dynamics and Function in Idesia polycarpa through Floral and Fruit Development
by Zhi Li, Qiupeng Yuan, Shasha Wang, Tao Zhang, Yanmei Wang, Qifei Cai, Xiaodong Geng, Yi Yang, Chao Miao, Li Dai, Sohel Rana and Zhen Liu
Microorganisms 2024, 12(10), 2022; https://doi.org/10.3390/microorganisms12102022 - 6 Oct 2024
Viewed by 404
Abstract
Male Idesia polycarpa, which display distinct morphological and physiological traits, exhibit greater adaptability to stressful environments than females. However, the connection between this adaptability and rhizosphere processes remains unclear. Here, we investigate the differences in root bacterial community structures between male and [...] Read more.
Male Idesia polycarpa, which display distinct morphological and physiological traits, exhibit greater adaptability to stressful environments than females. However, the connection between this adaptability and rhizosphere processes remains unclear. Here, we investigate the differences in root bacterial community structures between male and female plants at different developmental stages, identifying bacterial strains associated with plant sex through functional predictions. This study aims to inform the optimal allocation of male and female plants during cultivation and provide a theoretical basis for sex identification and breeding. Samples from seven-year-old male and female plants were collected during the flowering (May) and fruit ripening (October) stages. Rhizosphere nutrient content and bacterial diversity were analyzed using Illumina high-throughput sequencing technology. The results demonstrate that total nitrogen (TN), total carbon (TC), and available potassium (AK) varied between sexes at different times. No significant differences between male and female plants were observed in the Shannon, Simpson, and Chao1 indexes during the flowering period. However, the Chao1 and Shannon indexes were significantly higher at fruit maturity in male rather than female plants. The predominant phyla of rhizosphere bacteria were Pseudomonadota, Acidobacteriota, and Actinomycetes. Interestingly, from flowering to fruit ripening, the dominant phyla in both male and female plants shifted from Actinomycetes to Pseudomonadota. A significant correlation was observed between pH and AK and rhizosphere bacteria (p < 0.05), with metabolism being the main functional difference. This study provides preliminary insights into the functional predictions and analyses of bacteria associated with Idesia polycarpa. The above findings lay the groundwork for further investigation into the sex-specific differences in microbial flora across different developmental stages, elucidating the mechanisms underlying flora changes and offering theoretical support for the high-quality management of Idesia polycarpa. Full article
(This article belongs to the Special Issue Rhizosphere Microbial Community, 3rd Edition)
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<p>The soil properties of male and female plants of <span class="html-italic">Idesia polycarpa</span> at the flowering (May) and fruiting (October) stages. (<b>A</b>) Soil pH, (<b>B</b>) Total carbon, (<b>C</b>) Total nitrogen, (<b>D</b>) Available nitrogen, (<b>E</b>) Available phosphorus, and (<b>F</b>) Available potassium. The lowercase letters indicate significant differences (<span class="html-italic">p</span> &lt; 0.05). Data values represent the mean ± SE. Abbreviations: CS, female plant; XS, male plant.</p>
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<p>(<b>A</b>) Dilution curves of rhizosphere soil samples of <span class="html-italic">Idesia polycarpa</span> in different periods, (<b>B</b>) rhizosphere bacteria Venn diagram, (<b>C</b>) the number of OTUs of rhizosphere bacteria in different periods. Abbreviations: CS5, soil bacteria at the female flowering stage; CS10, soil bacteria at the female fruit ripening stage; XS5, soil bacteria at the male flowering stage; XS10, soil bacteria at the male fruit ripening stage.</p>
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<p>Soil bacteria in different periods in male and female plants of <span class="html-italic">Idesia polycarpa</span> represent (<b>A</b>) phylum level and (<b>B</b>) genus level. Abbreviations: CS5, soil bacteria at female flowering stage; XS5, soil bacteria at male flowering stage; CS10, soil bacteria at female fruit ripening stage; XS10, soil bacteria at male fruit ripening stage.</p>
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<p>LDA value distribution histogram of bacterial species in the rhizosphere of <span class="html-italic">Idesia polycarpa</span> in different periods. Abbreviations: CS5, soil bacteria at the female flowering stage; XS5, soil bacteria at the male flowering stage; CS10, soil bacteria at the female fruit ripening stage; XS10, soil bacteria at the male fruit ripening stage.</p>
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<p>Correlation analysis between soil factors and bacterial community structure. Abbreviations: pH, power of hydrogen; TC, total carbon; TN, Total nitrogen; AP, available phosphorus; AK, available potassium.</p>
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<p>Cluster analysis of soil nutrients and rhizosphere bacteria in different periods of male and female plants, through comparison with the KEGG database. Abbreviations: TC, total carbon; TN, total nitrogen; AK, available potassium; pH, power of hydrogen; AN, available nitrogen; AP, available phosphorus. A single asterisk represents * <span class="html-italic">p</span> &lt; 0.05; a double asterisk represents ** <span class="html-italic">p</span> &lt; 0.01; and a triple asterisk represents *** <span class="html-italic">p</span> &lt; 0.001 significance value.</p>
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<p>Differences in metabolic sub-functions between male and female plants at different stages (Cc-e: cellular community—eukaryotes, Cc-p: cellular community—prokaryotes, Dar: development and regeneration. Ds: digestive system, Eamd: endocrine and metabolic disease, Gbam: glycan biosynthesis and metabolism, Ss: sensory system, St: signal transduction). The lowercase letters indicate significant differences (<span class="html-italic">p</span> &lt; 0.05) among each cellular community’s female and male flowering and fruiting stages. Data values represent the mean ± SE. Abbreviations: CS5, soil bacteria at the female flowering stage; XS5, soil bacteria at the male flowering stage; CS10, soil bacteria at the female fruit ripening stage; XS10, soil bacteria at the male fruit ripening stage.</p>
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20 pages, 2554 KiB  
Article
Comprehensive Evaluation and Selection of Cardamom (Elettaria cardamomum (L.) Maton) Germplasm Using Morphological Traits
by Martha Patricia Herrera-González, Alejandra Zamora-Jerez, Rolando Cifuentes-Velasquez, Luis Andrés Arévalo-Rodríguez and Santiago Pereira-Lorenzo
Plants 2024, 13(19), 2786; https://doi.org/10.3390/plants13192786 - 4 Oct 2024
Viewed by 402
Abstract
Cardamom (Elettaria cardamomum (L.) Maton) plays a crucial role in Guatemala’s agriculture, supporting local families and covering 169,429.29 ha (making it the world’s leading producer). Since its introduction to Guatemala in 1910, limited research has focused on unraveling the diversity and defining [...] Read more.
Cardamom (Elettaria cardamomum (L.) Maton) plays a crucial role in Guatemala’s agriculture, supporting local families and covering 169,429.29 ha (making it the world’s leading producer). Since its introduction to Guatemala in 1910, limited research has focused on unraveling the diversity and defining morphological traits critical for selecting excellent accessions. In this study, we examined 17 morphological traits across 288 accessions to identify key features associated with the germplasm. The comprehensive analysis employed principal component analysis, a morphological composite value (F-value), linear regression, and hierarchical clustering. The Shannon–Wiener diversity index ranged from 0.10 to 2.02, indicating the variation in diversity among traits. Principal component analysis and hierarchical clustering revealed six distinct germplasm groups. The comprehensive analysis facilitated the selection of 14 excellent accessions, and the regression equation incorporating criteria such as plant height, capsule color, panicle number per plant, panicle length, rhizome color, cluster number per panicle, cluster node length, and capsule number per cluster to identify cardamom germplasm. To develop a conservation strategy for the two putative foreign varieties (‘Malabar’ and ‘Mysore’/’Vazhukka’) introduced in Guatemala based on plant height, another 12 accessions were selected with a second comprehensive evaluation. This information offers insights into cardamom diversity for informed selection enhancing national utilization, productivity, and conservation. Full article
(This article belongs to the Section Plant Genetic Resources)
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Graphical abstract

Graphical abstract
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<p>Hierarchical cluster analyses using the 3 PCs of the 17 morphological traits of 288 cardamom accessions from Guatemala.</p>
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<p>Selected cardamom accessions (1–14) classified into clusters (1–6) through hierarchical cluster analysis, utilizing the 3 principal components (PCs) derived from 17 morphological traits across 288 cardamom accessions.</p>
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<p>Principal component analysis (PC1–3) estimated with 17 morphological traits on 288 cardamom accessions classified in two putative foreign cultivars (‘Malabar’ and ‘Mysore’/’Vazhukka’) introduced in Guatemala according to (<b>a</b>) plant height and (<b>b</b>) color of the capsule.</p>
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<p>Collection sites of a sample of 288 accessions in the cardamom production regions at the Northern Transversal Strip of Guatemala.</p>
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24 pages, 1012 KiB  
Article
Bias in O-Information Estimation
by Johanna Gehlen, Jie Li, Cillian Hourican, Stavroula Tassi, Pashupati P. Mishra, Terho Lehtimäki, Mika Kähönen, Olli Raitakari, Jos A. Bosch and Rick Quax
Entropy 2024, 26(10), 837; https://doi.org/10.3390/e26100837 - 30 Sep 2024
Viewed by 394
Abstract
Higher-order relationships are a central concept in the science of complex systems. A popular method of attempting to estimate the higher-order relationships of synergy and redundancy from data is through the O-information. It is an information–theoretic measure composed of Shannon entropy terms that [...] Read more.
Higher-order relationships are a central concept in the science of complex systems. A popular method of attempting to estimate the higher-order relationships of synergy and redundancy from data is through the O-information. It is an information–theoretic measure composed of Shannon entropy terms that quantifies the balance between redundancy and synergy in a system. However, bias is not yet taken into account in the estimation of the O-information of discrete variables. In this paper, we explain where this bias comes from and explore it for fully synergistic, fully redundant, and fully independent simulated systems of n=3 variables. Specifically, we explore how the sample size and number of bins affect the bias in the O-information estimation. The main finding is that the O-information of independent systems is severely biased towards synergy if the sample size is smaller than the number of jointly possible observations. This could mean that triplets identified as highly synergistic may in fact be close to independent. A bias approximation based on the Miller–Maddow method is derived for the O-information. We find that for systems of n=3 variables the bias approximation can partially correct for the bias. However, simulations of fully independent systems are still required as null models to provide a benchmark of the bias of the O-information. Full article
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<p>The mean naive O-information estimation <math display="inline"><semantics> <mover> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mo>¯</mo> </mover> </semantics></math> per <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo>)</mo> </mrow> </semantics></math> combination over the 30 trials. This analysis is performed for the fully redundant triplet (<b>left</b> panel), the triplet of independent variables (<b>middle</b> panel), and the fully synergistic triplet (<b>right</b> panel).</p>
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<p>The bias <math display="inline"><semantics> <mi>δ</mi> </semantics></math> of the naive O-information estimation per <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo>)</mo> </mrow> </semantics></math> combination. Boundary lines indicate the theoretically minimum sample size, <span class="html-italic">N</span>, needed to observe the number of bins, <span class="html-italic">K</span>, as well as the joint bin combinations <math display="inline"><semantics> <msup> <mi>K</mi> <mrow> <mo>−</mo> <mi>j</mi> </mrow> </msup> </semantics></math> and <math display="inline"><semantics> <msup> <mi>K</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msup> </semantics></math>.</p>
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<p>The standard deviation of the naive O-information estimations <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mn>30</mn> </msub> </mrow> </semantics></math> per <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo>)</mo> </mrow> </semantics></math> combination over the 30 trials, for each of the three simulated systems.</p>
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<p>The difference in an estimation error between the naive and bias-corrected O-information estimation (<math display="inline"><semantics> <msup> <mi>ε</mi> <mo>′</mo> </msup> </semantics></math>). A negative value (blue) implies that <math display="inline"><semantics> <mover> <msub> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>B</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>¯</mo> </mover> </semantics></math> is more accurate than the naive estimation <math display="inline"><semantics> <mover> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mo>¯</mo> </mover> </semantics></math>. A positive value (red) implies that the naive estimation is more accurate.</p>
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<p>Behavior of the true O-information <math display="inline"><semantics> <mi mathvariant="sans-serif">Ω</mi> </semantics></math>, the mean naive O-information estimation <math display="inline"><semantics> <mover> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mo>¯</mo> </mover> </semantics></math>, and the bias-corrected O-information estimation <math display="inline"><semantics> <mover> <msub> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>B</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>¯</mo> </mover> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> bins.</p>
Full article ">Figure 6
<p>Behavior of the true O-information <math display="inline"><semantics> <mi mathvariant="sans-serif">Ω</mi> </semantics></math>, the mean naive O-information estimation <math display="inline"><semantics> <mover> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mo>¯</mo> </mover> </semantics></math>, and the mean bias-corrected O-information estimation <math display="inline"><semantics> <mover> <msub> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>B</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>¯</mo> </mover> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> bins.</p>
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<p>Distribution of the naive estimated O-information <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> and the bias-corrected estimated O-information <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>B</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> </msub> </semantics></math> for all triplets in the feature selected dataset. The dashed lines of <math display="inline"><semantics> <mrow> <mover> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mo>¯</mo> </mover> <mo>=</mo> <mo>−</mo> <mn>0.377</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mover> <msub> <mover accent="true"> <mi mathvariant="sans-serif">Ω</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>B</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>¯</mo> </mover> <mo>=</mo> <mo>−</mo> <mn>0.142</mn> </mrow> </semantics></math>, respectively, indicate the mean naive O-information estimation and the mean bias-corrected O-information estimation of the simulated independent triplets. The darker shaded distributions around the dashed lines are the distributions of the simulated O-information estimations.</p>
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23 pages, 6755 KiB  
Article
Neural Dynamics Associated with Biological Variation in Normal Human Brain Regions
by Natalí Guisande, Osvaldo A. Rosso and Fernando Montani
Entropy 2024, 26(10), 828; https://doi.org/10.3390/e26100828 - 29 Sep 2024
Viewed by 644
Abstract
The processes involved in encoding and decoding signals in the human brain are a continually studied topic, as neuronal information flow involves complex nonlinear dynamics. This study examines awake human intracranial electroencephalography (iEEG) data from normal brain regions to explore how biological sex [...] Read more.
The processes involved in encoding and decoding signals in the human brain are a continually studied topic, as neuronal information flow involves complex nonlinear dynamics. This study examines awake human intracranial electroencephalography (iEEG) data from normal brain regions to explore how biological sex influences these dynamics. The iEEG data were analyzed using permutation entropy and statistical complexity in the time domain and power spectrum calculations in the frequency domain. The Bandt and Pompe method was used to assess time series causality by associating probability distributions based on ordinal patterns with the signals. Due to the invasive nature of data acquisition, the study encountered limitations such as small sample sizes and potential sources of error. Nevertheless, the high spatial resolution of iEEG allows detailed analysis and comparison of specific brain regions. The results reveal differences between sexes in brain regions, observed through power spectrum, entropy, and complexity analyses. Significant differences were found in the left supramarginal gyrus, posterior cingulate, supplementary motor cortex, middle temporal gyrus, and right superior temporal gyrus. This study emphasizes the importance of considering sex as a biological variable in brain dynamics research, which is essential for improving the diagnosis and treatment of neurological and psychiatric disorders. Full article
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Figure 1

Figure 1
<p><b>Schema of the analyzed region locations along with the corresponding electrodes used.</b> The yellow area represents the region of interest, and the red dots indicate the positions of the electrodes.</p>
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<p><b>Power spectral density (PSD) and complexity-entropy causality plane of the superior parietal lobule region in the left hemisphere.</b> (<b>A</b>,<b>B</b>) PSD of female and male patients, respectively, with the solid line representing the median and the shaded area representing the interquartile range (IQR). (<b>C</b>) Median PSD of females and males. (<b>D</b>) Complexity-entropy causality plane using an embedding dimension <math display="inline"><semantics> <mrow> <mi>D</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>6</mn> </mrow> </semantics></math>, a time delay <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and a temporal window of 15 s for females and males. The boxplot shows the median, IQR, outliers, and the notch, which corresponds to the 95% confidence interval of the median.</p>
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<p><b>Power spectral density (PSD) and complexity-entropy causality plane of the supramarginal gyrus region in the left hemisphere.</b> (<b>A</b>,<b>B</b>) PSD of female and male patients, respectively, with the solid line representing the median and the shaded area representing the interquartile range (IQR). (<b>C</b>) Median PSD of females and males. (<b>D</b>) Complexity-entropy causality plane using an embedding dimension <math display="inline"><semantics> <mrow> <mi>D</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>6</mn> </mrow> </semantics></math>, a time delay <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and a temporal window of 15 s for females and males. The boxplot shows the median, IQR, outliers, and the notch, which corresponds to the 95% confidence interval of the median.</p>
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<p><b>Power spectral density (PSD) and complexity-entropy causality plane of the precuneus region in the left hemisphere.</b> (<b>A</b>,<b>B</b>) PSD of female and male patients, respectively, with the solid line representing the median and the shaded area representing the interquartile range (IQR). (<b>C</b>) Median PSD of females and males. (<b>D</b>) Complexity-entropy causality plane using an embedding dimension <math display="inline"><semantics> <mrow> <mi>D</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>6</mn> </mrow> </semantics></math>, a time delay <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and a temporal window of 15 s for females and males. The boxplot shows the median, IQR, outliers, and the notch, which corresponds to the 95% confidence interval of the median.</p>
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<p><b>Power spectral density (PSD) and complexity-entropy causality plane of the posterior cingulate region in the left hemisphere.</b> (<b>A</b>,<b>B</b>) PSD of female and male patients, respectively, with the solid line representing the median and the shaded area representing the interquartile range (IQR). (<b>C</b>) Median PSD of females and males. (<b>D</b>) Complexity-entropy causality plane using an embedding dimension <math display="inline"><semantics> <mrow> <mi>D</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>6</mn> </mrow> </semantics></math>, a time delay <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and a temporal window of 15 s for females and males. The boxplot shows the median, IQR, outliers, and the notch, which corresponds to the 95% confidence interval of the median.</p>
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<p><b>Power spectral density (PSD) and complexity-entropy causality plane of the supplementary motor cortex in the left hemisphere.</b> (<b>A</b>,<b>B</b>) PSD of female and male patients, respectively, with the solid line representing the median and the shaded area representing the interquartile range (IQR). (<b>C</b>) Median PSD of females and males. (<b>D</b>) Complexity-entropy causality plane using an embedding dimension <math display="inline"><semantics> <mrow> <mi>D</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>6</mn> </mrow> </semantics></math>, a time delay <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and a temporal window of 15 s for females and males. The boxplot shows the median, IQR, outliers, and the notch, which corresponds to the 95% confidence interval of the median.</p>
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<p><b>Power spectral density (PSD) and complexity-entropy causality plane of the triangular part of the inferior frontal gyrus in the left hemisphere.</b> (<b>A</b>,<b>B</b>) PSD of female and male patients, respectively, with the solid line representing the median and the shaded area representing the interquartile range (IQR). (<b>C</b>) Median PSD of females and males. (<b>D</b>) Complexity-entropy causality plane using an embedding dimension <math display="inline"><semantics> <mrow> <mi>D</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>6</mn> </mrow> </semantics></math>, a time delay <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and a temporal window of 15 s for females and males. The boxplot shows the median, IQR, outliers, and the notch, which corresponds to the 95% confidence interval of the median.</p>
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<p><b>Power spectral density (PSD) and complexity-entropy causality plane of the middle temporal gyrus in the left hemisphere.</b> (<b>A</b>,<b>B</b>) PSD of female and male patients, respectively, with the solid line representing the median and the shaded area representing the interquartile range (IQR). (<b>C</b>) Median PSD of females and males. (<b>D</b>) Complexity-entropy causality plane using an embedding dimension <math display="inline"><semantics> <mrow> <mi>D</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>6</mn> </mrow> </semantics></math>, a time delay <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and a temporal window of 15 s for females and males. The boxplot shows the median, IQR, outliers, and the notch, which corresponds to the 95% confidence interval of the median.</p>
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<p><b>Power spectral density (PSD) and complexity-entropy causality plane of the superior temporal gyrus in the right hemisphere.</b> (<b>A</b>,<b>B</b>) PSD of female and male patients, respectively, with the solid line representing the median and the shaded area representing the interquartile range (IQR). (<b>C</b>) Median PSD of females and males. (<b>D</b>) Complexity-entropy causality plane using an embedding dimension <math display="inline"><semantics> <mrow> <mi>D</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>6</mn> </mrow> </semantics></math>, a time delay <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and a temporal window of 15 s for females and males. The boxplot shows the median, IQR, outliers, and the notch, which corresponds to the 95% confidence interval of the median.</p>
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30 pages, 5446 KiB  
Article
On the Exploration of Quantum Polar Stabilizer Codes and Quantum Stabilizer Codes with High Coding Rate
by Zhengzhong Yi, Zhipeng Liang, Yulin Wu and Xuan Wang
Entropy 2024, 26(10), 818; https://doi.org/10.3390/e26100818 - 25 Sep 2024
Viewed by 452
Abstract
Inspired by classical polar codes, whose coding rate can asymptotically achieve the Shannon capacity, researchers are trying to find their analogs in the quantum information field, which are called quantum polar codes. However, no one has designed a quantum polar coding scheme that [...] Read more.
Inspired by classical polar codes, whose coding rate can asymptotically achieve the Shannon capacity, researchers are trying to find their analogs in the quantum information field, which are called quantum polar codes. However, no one has designed a quantum polar coding scheme that applies to quantum computing yet. There are two intuitions in previous research. The first is that directly converting classical polar coding circuits to quantum ones will produce the polarization phenomenon of a pure quantum channel, which has been proved in our previous work. The second is that based on this quantum polarization phenomenon, one can design a quantum polar coding scheme that applies to quantum computing. There are several previous work following the second intuition, none of which has been verified by experiments. In this paper, we follow the second intuition and propose a more reasonable quantum polar stabilizer code construction algorithm than any previous ones by using the theory of stabilizer codes. Unfortunately, simulation experiments show that even the stabilizer codes obtained from this more reasonable construction algorithm do not work, which implies that the second intuition leads to a dead end. Based on the analysis of why the second intuition does not work, we provide a possible future direction for designing quantum stabilizer codes with a high coding rate by borrowing the idea of classical polar codes. Following this direction, we find a class of quantum stabilizer codes with a coding rate of 0.5, which can correct two of the Pauli errors. Full article
(This article belongs to the Special Issue Quantum Computing in the NISQ Era)
Show Figures

Figure 1

Figure 1
<p>Classical channel polarization. <math display="inline"><semantics> <msub> <mi>R</mi> <mi>N</mi> </msub> </semantics></math> is the reverse shuffle operation. Two independent copies of <math display="inline"><semantics> <msub> <mi>W</mi> <mrow> <mi>N</mi> <mo>/</mo> <mn>2</mn> </mrow> </msub> </semantics></math> are combined to produce the channel <math display="inline"><semantics> <msub> <mi>W</mi> <mi>N</mi> </msub> </semantics></math>. Please see ref. [<a href="#B11-entropy-26-00818" class="html-bibr">11</a>] for more detail.</p>
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<p>Quantum channel polarization circuits: (<b>a</b>) Two primal channel <math display="inline"><semantics> <mi mathvariant="script">E</mi> </semantics></math> combines to form channel <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mn>2</mn> </msub> </semantics></math>. (<b>b</b>) Two <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mn>2</mn> </msub> </semantics></math> combines to form channel <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mn>4</mn> </msub> </semantics></math>. (<b>c</b>) Two <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mrow> <mi>N</mi> <mo>/</mo> <mn>2</mn> </mrow> </msub> </semantics></math> combines to form channel <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mi>N</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>R</mi> <mi>N</mi> </msub> </semantics></math> is the reverse shuffle operator [<a href="#B21-entropy-26-00818" class="html-bibr">21</a>].</p>
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<p>Transformation of Pauli <span class="html-italic">X</span> operators and Pauli <span class="html-italic">Z</span> operators under conjugation by CNOT gate.</p>
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<p>The coding rate of CA and BS algorithms with different physical qubit error rates and code lengths <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> </semantics></math>. (<b>a</b>) Physical error rate ranges from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>b</b>) Physical error rate ranges from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>c</b>) Physical error rate ranges from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
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<p>The LER with unreliable frozen qubits and table-look-up decoder. “Logical error rate” represents <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> and “Slq Logical error rate” represents <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math>. (<b>a</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>b</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>c</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>d</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>e</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>f</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>g</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>h</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>i</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>The LER with unreliable frozen qubits and bit-flip decoder. “Logical error rate” represents <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> and “Slq Logical error rate” represents <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math>. (<b>a</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>b</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>c</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>d</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>e</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>f</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>g</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>h</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>i</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 7
<p>The performance of table-look-up decoder and bit-flip decoder in the range of <span class="html-italic">p</span> from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. The code length is set to <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>2</mn> <mn>6</mn> </msup> </mrow> </semantics></math>. “Logical error rate” represents <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>. (<b>a</b>) Physical error rate ranges from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>b</b>) Physical error rate ranges from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>c</b>) Physical error rate ranges from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>The performance of table-look-up decoder under CA algorithm with different <math display="inline"><semantics> <msub> <mi>t</mi> <mrow> <mi>e</mi> <mi>r</mi> <mi>r</mi> <mi>o</mi> <mi>r</mi> </mrow> </msub> </semantics></math>. “Logical error rate” represents <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>. (<b>a</b>) Physical error rate ranges from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>b</b>) Physical error rate ranges from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 9
<p>The LER with reliable frozen qubits and table-look-up decoder. “Logical error rate” represents <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> and “Slq Logical error rate” represents <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math>. (<b>a</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>b</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>c</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>d</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>e</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>f</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>g</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>h</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>i</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 10
<p>The LER with reliable frozen qubits and bit-flip decoder. “Logical error rate” represents <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> and “Slq Logical error rate” represents <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math>. (<b>a</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>b</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>c</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>d</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>e</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>f</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>g</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>l</mi> <mi>o</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA and BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>h</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by CA algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>i</b>) The <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>E</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math> of QPSCs constructed by BS algorithms with physical error rate ranging from <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
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<p>The effective code distance of CA and BS algorithms with different physical qubit error rates and code lengths <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> </semantics></math>. (<b>a</b>) The effective code distance of QPSCs constructed by CA algorithms. (<b>b</b>) The effective code distance of QPSCs constructed by BS algorithms.</p>
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<p>Coordinate channels and decoding channels: (<b>a</b>) Classical coordinate channels. The input of classical coordinate channel <math display="inline"><semantics> <msubsup> <mi>W</mi> <mi>N</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math> is <math display="inline"><semantics> <msub> <mi>u</mi> <mi>i</mi> </msub> </semantics></math>, and its output is <math display="inline"><semantics> <mrow> <msubsup> <mi>y</mi> <mn>1</mn> <mi>N</mi> </msubsup> <mo>,</mo> <msubsup> <mi>u</mi> <mn>1</mn> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </semantics></math>. (<b>b</b>) Classical decoding channels. The input of decoding channel <math display="inline"><semantics> <msubsup> <mi>D</mi> <mi>N</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math> is <math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>u</mi> <mo stretchy="false">^</mo> </mover> <mn>1</mn> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mn>1</mn> <mi>N</mi> </msubsup> </mrow> </semantics></math> and the output is estimated <math display="inline"><semantics> <msub> <mover accent="true"> <mi>u</mi> <mo stretchy="false">^</mo> </mover> <mi>i</mi> </msub> </semantics></math>. (<b>c</b>) Quantum coordinate channels. The input of quantum coordinate channel <math display="inline"><semantics> <msubsup> <mi mathvariant="script">E</mi> <mi>N</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math> is <math display="inline"><semantics> <msup> <mi>ρ</mi> <msub> <mi>Q</mi> <mi>i</mi> </msub> </msup> </semantics></math>, and its output is <math display="inline"><semantics> <msup> <mi>ρ</mi> <mrow> <msubsup> <mi>Y</mi> <mn>1</mn> <mi>N</mi> </msubsup> <mo>,</mo> <msubsup> <mi>R</mi> <mn>1</mn> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </msup> </semantics></math>. (<b>d</b>) Quantum decoding channel. The input of decoding channel is an error syndrome and the output is the most likely error.</p>
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<p>The Tanner graph of 3-bit-flip code.</p>
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<p>Quantum stabilizer codes by the recursive expansion of Tanner graph. The arrow means the corresponding qubit it starts from will join in the corresponding stabilizer it ends with. The corresponding stabilizers and logical operators are shown in <a href="#entropy-26-00818-t001" class="html-table">Table 1</a>: (<b>a</b>) The Tanner graph <math display="inline"><semantics> <msub> <mi>G</mi> <mn>1</mn> </msub> </semantics></math> used for recursive expansion. (<b>b</b>) The expanded Tanner graph <math display="inline"><semantics> <msub> <mi>G</mi> <mn>2</mn> </msub> </semantics></math> by recursive expansion of two <math display="inline"><semantics> <msub> <mi>G</mi> <mn>1</mn> </msub> </semantics></math>. (<b>c</b>) The expanded Tanner graph <math display="inline"><semantics> <msub> <mi>G</mi> <mn>3</mn> </msub> </semantics></math> by recursive expansion of two <math display="inline"><semantics> <msub> <mi>G</mi> <mn>2</mn> </msub> </semantics></math>. (<b>d</b>) The expanded Tanner graph <math display="inline"><semantics> <msub> <mi>G</mi> <mi>n</mi> </msub> </semantics></math> by recursive expansion of two <math display="inline"><semantics> <msub> <mi>G</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </semantics></math>.</p>
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<p>The LER with table-look-up decoder.</p>
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<p>The decoding accuracy under different <math display="inline"><semantics> <msub> <mi>t</mi> <mrow> <mi>e</mi> <mi>r</mi> <mi>r</mi> <mi>o</mi> <mi>r</mi> </mrow> </msub> </semantics></math>.</p>
Full article ">Figure A1
<p>The flow chart corresponds to Algorithm 3.</p>
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15 pages, 1022 KiB  
Article
The Genetic Diversity of 69 Widely Used Chinese Sorghum Hybrids Released between the 1970s and 2010s
by Haisheng Yan, Na Lv, Feng Yin, Yubin Wang, Hao Niu, Xin Lv, Jianqiang Chu, Fangfang Fan, Lan Ju, Jizhen Yu, Fuyao Zhang and Junai Ping
Agronomy 2024, 14(10), 2180; https://doi.org/10.3390/agronomy14102180 - 24 Sep 2024
Viewed by 464
Abstract
Sorghum has a long history of cultivation in China. In this study, we aimed to clarify the genetic relationships and genetic variation trends in widely used Chinese sorghum hybrids which were released from the 1970s to 2010s and attempted to analyze the changes [...] Read more.
Sorghum has a long history of cultivation in China. In this study, we aimed to clarify the genetic relationships and genetic variation trends in widely used Chinese sorghum hybrids which were released from the 1970s to 2010s and attempted to analyze the changes in sorghum breeding. A total of 257 alleles were detected by 51 polymorphic SSR markers among 69 widely used hybrids; an average of 5.04 alleles were detected by each marker. The average Shannon’s index and polymorphism information content (PIC) of markers were 1.39 and 0.70, respectively. Nei’s genetic diversity index continuously increased in four different breeding development stages (1973–1982, 1983–1992, 1993–2002, and 2003–2014). Genetic diversity gradually increased among the sorghum hybrids. Genetic similarity coefficients in the four breeding development stages first showed an increasing trend, and then a decreasing trend, finally stabilizing with an average value of 0.65. The genetic similarity changes in hybrids in early and late maturing areas were consistent at different breeding development stages. The genetic similarity coefficients in late maturing areas were constantly higher than those in the early maturing areas. This is related to China’s creative utilization of A2 cytoplasmic male sterile materials in the 1990s. A cluster analysis determined that 69 hybrids were divided into two groups, A and B. Group A could be further subdivided into four subgroups. These findings could provide a reference for parental selection and hybrid breeding in sorghum improvement programs. Full article
(This article belongs to the Section Crop Breeding and Genetics)
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<p>Dendrogram representing genetic relationships among 69 hybrid sorghum hybrids based on results of SSR analysis.</p>
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15 pages, 1572 KiB  
Article
Exploring Sampling Strategies and Genetic Diversity Analysis of Red Beet Germplasm Resources Using SSR Markers
by Xiangjia Wu, Zhi Pi, Shengnan Li and Zedong Wu
Horticulturae 2024, 10(9), 1008; https://doi.org/10.3390/horticulturae10091008 - 23 Sep 2024
Viewed by 491
Abstract
By using 14 SSR primer pairs, we here analyzed and compared the amplification results of 534 DNA samples from six red sugar beet germplasm resources under three treatments. These data were used to explore the sampling strategy for the aforementioned resources. Based on [...] Read more.
By using 14 SSR primer pairs, we here analyzed and compared the amplification results of 534 DNA samples from six red sugar beet germplasm resources under three treatments. These data were used to explore the sampling strategy for the aforementioned resources. Based on the sampling strategy results, 21 SSR primer pairs were used to evaluate the genetic diversity of 47 red sugar beet germplasm resources. The six population genetic parameters used for exploring the sampling strategy unveiled that individual plants within the population had a relatively large genetic distance. The genetic parameters Ne, I, and Nei’s of the randomly mixed sampling samples increased rapidly between 10 and 30 plants before decreasing. Therefore, when SSR technology was used to analyze the genetic diversity of the red sugar beet germplasm resources, the optimal sampling gradient for each population was the adoption of a random single-plant mixed sampling sample of no less than 10 plants and no more than 30 plants. The 21 SSR primer pairs were used to detect genetic diversity in 30 random mixed samples of 47 resources. The average polymorphic information content (PIC) was 0.5738, the average number of observed alleles (Na) was 4.1905, the average number of effective alleles (Ne) was 2.8962, the average Shannon’s information index (I) was 1.1299, the average expected heterozygosity (Nei’s) was 0.6127, and the average expected heterozygosity (He) was 0.6127. The genetic distance of the 47 germplasm resources ranged from 0.0225 to 0.551 (average: 0.316). According to the population structure analysis, the most suitable K value was six, which indicated the presence of six populations. Based on the clustering analysis results, the 47 germplasm resources were segregated into six groups, with obvious clustering and some germplasm resources noted for having groups with close genetic relationships. We here established a more accurate and scientific sampling strategy for analyzing the genetic diversity of red sugar beet germplasm resources by using SSR molecular markers. The findings provide a reference for collecting and preserving red sugar beet germplasms and protecting their genetic diversity. Full article
(This article belongs to the Section Genetics, Genomics, Breeding, and Biotechnology (G2B2))
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<p>Amplification band map of primer TC94 on 47 red sugar beet germplasm resources.</p>
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<p>A cluster diagram of 47 red sugar beet varieties (lines) based on SSR markers. Note: The numbers correspond to the names of the tested varieties (lines) in <a href="#horticulturae-10-01008-t001" class="html-table">Table 1</a>;A–F represent six different categories respectively.</p>
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<p>The relationship between the optimal number of taxa (K) and the inferred value (ΔK).</p>
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<p>Genetic structure of 47 red beet varieties (lines) based on SSR markers. Note: Six different colors represent six different groups; The size of the color distribution area in the figure represents the proportion of different groups.</p>
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17 pages, 10769 KiB  
Article
Enhancing In Situ Conservation of Crop Wild Relatives for Food and Agriculture in Lithuania
by Juozas Labokas, Mantas Lisajevičius, Domas Uogintas and Birutė Karpavičienė
Agronomy 2024, 14(9), 2126; https://doi.org/10.3390/agronomy14092126 - 18 Sep 2024
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Abstract
The crop and crop wild relative (CWR) checklist of Lithuania was created containing 2630 taxa. The checklist comprises 1384 native taxa including archaeophytes and 1246 neophytes. In total, 699 taxa (26.6%) are defined for food and forage use. A list of 144 CWR [...] Read more.
The crop and crop wild relative (CWR) checklist of Lithuania was created containing 2630 taxa. The checklist comprises 1384 native taxa including archaeophytes and 1246 neophytes. In total, 699 taxa (26.6%) are defined for food and forage use. A list of 144 CWR priority species with 135 native species and archaeophytes and 9 naturalized species was generated. In total, 53 genera of food and forage species belonging to 15 families are represented by the priority CWR. Two approaches for CWR genetic reserve selection have been employed in this study: (1) CWR-targeted evaluation of preselected sites, including Natura 2000 sites, national protected areas, and other effective area-based conservation measures (OECMs), such as ancient hillfort sites and ecological protection zones of water bodies; (2) analysis of large georeferenced plant databases. Forty-five potential genetic reserve sites have been selected by the first approach covering 83 species or 57.6% of the national CWR priority list. With the second approach, the in situ CWR National Inventory database has been created by combining data from the Database of EU habitat mapping in Lithuania (BIGIS), Herbarium Database of the Nature Research Centre (BILAS), Lithuanian Vegetation Database (EU-LT-001), and Global Biodiversity Information Facility (GBIF). Hotspot analysis of CWR species richness and number of observations suggested that higher CWR diversity is more likely to be found in protected areas. However, Shannon diversity and Shannon equitability indices showed that the areas outside of the protected areas are also suitable for CWR genetic reserve establishment. Full article
(This article belongs to the Section Agroecology Innovation: Achieving System Resilience)
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<p>Distribution of 45 potential CWR genetic reserve sites on climate map of Lithuania [<a href="#B47-agronomy-14-02126" class="html-bibr">47</a>] established by the evaluation of preselected sites on state-owned land. For details on CWR genetic reserve sites, see the <a href="#app1-agronomy-14-02126" class="html-app">Supplementary Materials (Table S3)</a>.</p>
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<p>Species richness (numbers of species per grid cell 4 × 4 km) of priority CWR across Lithuania and locations of the preselected 45 potential CWR reserve sites.</p>
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<p>Species richness (numbers of species per grid cell 4 × 4 km) of priority CWR in protected areas of Lithuania including Natura 2000 network.</p>
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<p>Number of species observations (per grid cell 4 × 4 km) of priority CWR across Lithuania.</p>
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<p>Number of species observations (per grid cell 4 × 4 km) of priority CWR in protected areas of Lithuania including Natura 2000 network.</p>
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14 pages, 23115 KiB  
Article
Assessment of Population Genetic Diversity of Medicinal Meconopsis integrifolia (Maxim.) Franch. Using Newly Developed SSR Markers
by Jiahao Wu, Quanyin Yang, Wanyue Zhao, Xue Miao, Yuan Qin, Yan Qu and Ping Zheng
Plants 2024, 13(18), 2561; https://doi.org/10.3390/plants13182561 - 12 Sep 2024
Viewed by 391
Abstract
Meconopsis integrifolia is an endangered Tibetan medicinal plant with significant medicinal and ornamental value. Understanding its genetic diversity and structure is crucial for its sustainable utilization and effective conservation. Here, we develop a set of SSR markers based on transcriptome data to analyze [...] Read more.
Meconopsis integrifolia is an endangered Tibetan medicinal plant with significant medicinal and ornamental value. Understanding its genetic diversity and structure is crucial for its sustainable utilization and effective conservation. Here, we develop a set of SSR markers based on transcriptome data to analyze the genetic diversity and structure of 185 individuals from 16 populations of M. integrifolia. The results indicate that M. integrifolia exhibits relatively high genetic diversity at the species level (the percentage of polymorphic bands PPB = 91.67%, Nei’s genetic diversity index He = 0.2989, Shannon’s information index I = 0.4514) but limited genetic variation within populations (PPB = 12.08%, He = 0.0399, I = 0.0610). The genetic differentiation among populations is relatively high (the coefficient of gene differentiation GST = 0.6902), and AMOVA analysis indicates that 63.39% of the total variation occurs among populations. This suggests that maintaining a limited number of populations is insufficient to preserve the overall diversity of M. integrifolia. Different populations are categorized into four representative subclusters, but they do not cluster strictly according to geographical distribution. Limited gene flow (Nm = 0.2244) is likely the main reason for the high differentiation among these populations. Limited seed and pollen dispersal abilities, along with habitat fragmentation, may explain the restricted gene flow among populations, highlighting the necessity of conserving as many populations in the wild as possible. Full article
(This article belongs to the Section Plant Genetics, Genomics and Biotechnology)
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<p>Habitat (<b>A</b>), whole plant morphology (<b>B</b>), and flower morphology (<b>C</b>) of <span class="html-italic">M. integrifolia</span> in the wild.</p>
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<p>Characterizations of SSRs identified from transcriptome datasets obtained in our previous work of <span class="html-italic">Meconopsis</span> “Lingholm”. (<b>A</b>) Distribution of the length of repeats in SSR loci. (<b>B</b>) Distribution of the number of repeats in SSR loci. (<b>C</b>) Type distribution of SSRs identified in the assembled <span class="html-italic">Meconopsis</span> “Lingholm” unigenes.</p>
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<p>PCR amplification results using primers P69 and P29 in <span class="html-italic">M. integrifolia</span>. The gel image shows the amplification products of individuals from different populations, with bands separated by markers. The full names corresponding to the population abbreviations in the figure are provided in <a href="#plants-13-02561-t002" class="html-table">Table 2</a>.</p>
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<p>Genetic distance (below diagonal) and genetic similarity (above diagonal) among 16 populations of <span class="html-italic">M. integrifolia</span> based on SSR analysis. “****” indicates the comparison of populations with themselves, which naturally results in perfect similarity and theoretically a genetic distance of zero.</p>
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<p>UPGMA dendrogram based on Nei’s genetic diversity coefficient among 16 populations of <span class="html-italic">M. integrifolia</span> using SSR marker analysis. Bootstrap analysis was performed with 1000 replicates, and the corresponding bootstrap values (%) were indicated.</p>
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<p>PCA cluster analysis of 185 individuals from 16 populations of <span class="html-italic">M. integrifolia</span>.</p>
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<p>Location distributions of the 16 populations of <span class="html-italic">M. integrifolia</span> sampled for this study. Six of the sixteen populations were collected from Tibet, four from Yunnan Province, and three each from Qinghai Province and Sichuan Province. The number of samples collected from each population is also indicated below the corresponding population name. Details of the populations are provided in <a href="#app1-plants-13-02561" class="html-app">Table S1</a>.</p>
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16 pages, 1490 KiB  
Article
Genetic Diversity and Population Genetic Structure of Jatropha curcas L. Accessions from Different Provenances Revealed by Amplified Fragment-Length Polymorphism and Inter-Simple Sequence Repeat Markers
by Guoye Guo, Lin Tang and Ying Xu
Forests 2024, 15(9), 1575; https://doi.org/10.3390/f15091575 - 8 Sep 2024
Viewed by 532
Abstract
The genetic diversity and structure of 17 populations of J. curcas, including 92 accessions from different provenances (tropical and subtropical), were investigated and effectively evaluated using twelve inter-simple sequence repeats (ISSRs) and seven pairs of florescence-amplified fragment-length polymorphism (AFLP) primers. Genetic diversity, [...] Read more.
The genetic diversity and structure of 17 populations of J. curcas, including 92 accessions from different provenances (tropical and subtropical), were investigated and effectively evaluated using twelve inter-simple sequence repeats (ISSRs) and seven pairs of florescence-amplified fragment-length polymorphism (AFLP) primers. Genetic diversity, at the overall level among populations of J. curcas based on the ISSR markers, showed that the observed number of alleles (Na) was 1.593, the effective number of alleles (Ne) was 1.330, Nei’s gene diversity (H) was 0.200, Shannon’s information index (I) was 0.303, and the percentage of polymorphic loci was 59.29%, indicating moderate genetic diversity between and within the different populations of J. curcas. Based on the genetic diversity analysis of AFLP markers, there were 1.464 (Na) and 1.216 (Ne) alleles, Nei’s gene diversity (H) was 0.132, Shannon’s information index (I) was 0.204, and the percentage of polymorphic loci was 46.40%. The AMOVA analysis showed that this large variance was due to differences within the populations, with genetic distinctions and limited gene flow among those from varied regions. The 17 populations were clustered into five main groups via UPGMA clustering analysis based on Nei’s genetic distance, and the genetic relationships among the populations exhibited no significant correlations with geographical provenances. The genetic variation among Chinese populations of J. curcas distributed in dry-hot valley areas was remarkable, and the American germplasm presented with distinct genetic differentiation. Full article
(This article belongs to the Special Issue Genetic Diversity and Gene Analysis in Forest Tree Breeding)
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<p>UPGMA dendrogram based on Nei’s (1978) unbiased genetic distance among 17 <span class="html-italic">J. curcas</span> populations using ISSR markers. The scale represents Nei’s similarity coefficient. Bold letters A, B, C, D, and E represent clustering groups; different colored lines correspond to different groups.</p>
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<p>UPGMA dendrogram based on genetic distance among 17 <span class="html-italic">J. curcas</span> populations using AFLP markers. The scale represents Nei’s similarity coefficient. Bold letters A, B, C, D, and E represent clustering groups; different colored lines correspond to different groups.</p>
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<p>UPGMA dendrogram based on genetic distance among 17 <span class="html-italic">J. curcas</span> populations using combined ISSR and AFLP markers. The scale represents Nei’s similarity coefficient. Bold letters A, B, C, D, and E represent clustering groups; different colored lines correspond to different groups.</p>
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13 pages, 6804 KiB  
Article
Analysis of Phenotypic Trait Variation in Germplasm Resources of Lycium ruthenicum Murr.
by Rong Yang, Jinpu Li, Haiguang Huang, Xiuhua Wu, Riheng Wu and Yu’e Bai
Agronomy 2024, 14(9), 1930; https://doi.org/10.3390/agronomy14091930 - 28 Aug 2024
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Abstract
Exploring the phenotypic trait variation and diversity of Lycium ruthenicum germplasm resources can support selection, breeding, and genetic improvement, enhancing agricultural production. This study collected 213 wild Lycium ruthenicum seedlings from a resource nursery in Alxa League, Inner Mongolia. These seedlings originated from [...] Read more.
Exploring the phenotypic trait variation and diversity of Lycium ruthenicum germplasm resources can support selection, breeding, and genetic improvement, enhancing agricultural production. This study collected 213 wild Lycium ruthenicum seedlings from a resource nursery in Alxa League, Inner Mongolia. These seedlings originated from eight sources across four provinces. Using 11 pseudo-qualitative traits and 20 quantitative traits, the phenotypic variation of the germplasm was analyzed. The analysis involved the coefficient of variation, Shannon–Wiener index (H), Simpson’s genetic diversity index (D), principal component analysis, correlation analysis, and Q-type cluster analysis. The results showed that the variation range of 31 phenotypic traits across the 213 Lycium ruthenicum germplasm resources was 17.26% to 105.41%, with an average coefficient of variation of 39.85%. The H and D indexes ranged from 0.18 to 1.58 and 0.20 to 0.75, respectively. For the 11 pseudo-qualitative traits, the H and D ranges were 0.18 to 1.58 and 0.07 to 0.74, with average values of 0.77 and 0.42. For the quantitative traits, the H and D ranges were 0.54 to 1.49 and 0.25 to 0.75, with average values of 1.21 and 0.63. This indicates that Lycium ruthenicum germplasm resources exhibit significant phenotypic diversity, with quantitative traits showing higher diversity than pseudo-qualitative traits. Principal component analysis revealed that the cumulative variance contribution rate of the first 10 principal components was 74.03%, comprehensively reflecting the information of the 31 traits. Q-type cluster analysis grouped the 213 Lycium ruthenicum germplasm resources into six clusters, each with distinct phenotypic characteristics. This analysis also identified the trait characteristics and breeding value of each cluster. The results of this study provide valuable information on the genetic improvement, conservation, and evaluation of Lycium ruthenicum germplasm resources. Full article
(This article belongs to the Section Crop Breeding and Genetics)
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<p>Frequency distribution of pseudo-qualitative traits in 213 <span class="html-italic">Lycium ruthenicum</span>.</p>
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<p>Diversity of pseudo-qualitative traits in 213 <span class="html-italic">Lycium ruthenicum</span>.</p>
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<p>Correlation analysis of 31 phenotypic traits. NFBBOB stands for Number of Fruit-Bearing Branches on One-Year-Old Branches, and MNFCSBOB stands for Maximum Number of Flowers in Clusters on Short Branches of One-Year-Old Branches.</p>
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<p>Q-type cluster analysis of 213 <span class="html-italic">Lycium ruthenicum</span> Murr.</p>
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