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Some Combinatorial Aspects of Constructing Bipartite-Graph Codes

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Abstract

We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix contain at most one unit in common. These matrices are equivalent to n-regular bipartite graphs without 4-cycles, and therefore can be used for the construction of efficient bipartite-graph codes such that both the classes of its vertices are associated with local constraints. We significantly extend the region of parameters m, n for which there exist an n-regular bipartite graph with 2m vertices and without 4-cycles. In that way we essentially increase the region of lengths and rates of the corresponding bipartite-graph codes. Many new matrices are either circulant or consist of circulant submatrices: this provides code parity-check matrices consisting of circulant submatrices, and hence quasi-cyclic bipartite-graph codes with simple implementation.

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Authors and Affiliations

  1. Institute for Information Transmission Problems, Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 127994, Russian Federation

    Alexander A. Davydov

  2. Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123, Perugia, Italy

    Massimo Giulietti, Stefano Marcugini & Fernanda Pambianco

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  1. Alexander A. Davydov
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  2. Massimo Giulietti
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  3. Stefano Marcugini
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  4. Fernanda Pambianco
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Correspondence to Alexander A. Davydov.

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Davydov, A.A., Giulietti, M., Marcugini, S. et al. Some Combinatorial Aspects of Constructing Bipartite-Graph Codes. Graphs and Combinatorics 29, 187–212 (2013). https://doi.org/10.1007/s00373-011-1103-5

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  • DOI: https://doi.org/10.1007/s00373-011-1103-5

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