Abstract
Let X be a complete simplicial toric variety over a finite field with a split torus \(T_{X}\). For any matrix Q, we are interested in the subgroup \(Y_{Q}\) of \(T_{X}\) parameterized by the columns of Q. We give an algorithm for obtaining a basis for the unique lattice L whose lattice ideal \(I_{L}\) is \(I(Y_{Q})\). We also give two direct algorithmic methods to compute the order of \(Y_{Q}\), which is the length of the corresponding code \({{\mathcal {C}}}_{\mathbf{\alpha },Y_Q}\). We share procedures implementing them in \({\texttt {Macaulay2}}\). Finally, we give a lower bound for the minimum distance of \({{\mathcal {C}}}_{\mathbf{\alpha },Y_Q}\), taking advantage of the parametric description of the subgroup \(Y_Q\). As an application, we compute the main parameters of the toric codes on Hirzebruch surfaces \({\mathcal {H}}_{\ell }\) generalizing the corresponding result given by Hansen.
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Acknowledgements
The article is part of the first author’s PhD thesis under the supervision by the second author. She is grateful to Hacettepe University Mathematics Department for the scientific environment. The authors thank Oğuz Yayla for his valuable helps on Sect. 6. They also thank an anonymous referee for helpful comments and suggestions improving the presentation of the paper.
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The first author is supported by TÜBİTAK-2211, the second author is supported by TÜBİTAK Project No: 114F094.
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Baran, E., Şahin, M. On parameterized toric codes. AAECC 34, 443–467 (2023). https://doi.org/10.1007/s00200-021-00513-8
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DOI: https://doi.org/10.1007/s00200-021-00513-8
Keywords
- Evaluation code
- Toric variety
- Multigraded Hilbert function
- Vanishing ideal
- Parameterized code
- Lattice ideal