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Hulls of codes from incidence matrices of connected regular graphs

机译：连接正则图的入射矩阵的代码外壳

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摘要

The hulls of codes from the row span over F_p, for any prime p, of incidence matrices of connected k-regular graphs are examined, and the dimension of the hull is given in terms of the dimension of the row span of A + k I over F_I, where A is an adjacency matrix for the graph. If p = 2, for most classes of connected regular graphs with some further form of symmetry, it was shown by Dankelmann et al. (Des. Codes Cryptogr. 2012) that the hull is either {0} or has minimum weight at least 2k - 2. Here we show that if the graph is strongly regular with parameter set (n, k, X, μ), then, unless k is even and μ is odd, the binary hull is non-trivial, of minimum weight generally greater than 2k - 2, and we construct words of low weight in the hull; if it is even and μ is odd, we show that the binary hull is zero. Further, if a graph is the line graph of a k-regular graph, it ≥ 3, that has an l-cycle for some l ≥ 3, the binary hull is shown to be non-trivial with minimum weight at most 2l(k - 2). Properties of the p-ary hulls are also established.
机译：对于连接的k正则图的入射矩阵，对于任意素数p，检查了行跨度F_p上的代码的外壳，并根据A + k I的行跨度的尺寸给出了外壳的尺寸在F_I上，其中A是图的邻接矩阵。如果p = 2，则对于大多数具有某种其他形式对称性的正则图连接类别，Dankelmann等人证明了这一点。 （Des。Codes Cryptogr。2012）船体是{0}或最小权重至少为2k-2。在这里，我们表明，如果该图是带有参数集（n，k，X，μ）的强规则，则，除非k为偶数且μ为奇数，否则二进制船体不是平凡的，最小权重通常大于2k-2，我们在船体中构造了低权重的单词；如果为偶数且μ为奇数，则表明二进制外壳为零。此外，如果图是k正则图的线图，则它≥3，对于l≥3具有l周期，则二进制船体显示为非平凡的，最小权重最大为2l（k -2）。还建立了p形船体的属性。

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  • 来源
    《Designs, Codes and Crytography》 |2014年第2期|35-54|共20页
  • 作者

    D. Ghinelli; J. D. Key; T. P. McDonough;

  • 作者单位

    Dipartimento di Matematica, Universita di Roma 'La Sapienza', 00185 Rome, Italy;

    Department of Mathematics and Applied Mathematics, University of the Western Cape, 7535 Bellvillek South Africa;

    Institute of Mathematics and Physics, Aberystwyth University, Aberystwyth, Ceredigion SY23 3BZ, UK;

  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

    Incidence matrix; Graph; Code; Hull; Permutation decoding;

    机译：发病率矩阵;图形;码;船体置换解码;

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