Hulls of codes from incidence matrices of connected regular graphs

  • Authors:

  • D. Ghinelli;J. D. Key;T. P. Mcdonough

  • Affiliations:

  • Dipartimento di Matematica, Università di Roma `La Sapienza', Rome, Italy 00185;Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville, South Africa 7535;Institute of Mathematics and Physics, Aberystwyth University, Aberystwyth, UK SY23 3BZ

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2014

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Abstract

The hulls of codes from the row span over $${\mathbb{F}_p}$$ , 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 + kI over $${\mathbb{F}_p}$$ , 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驴2. Here we show that if the graph is strongly regular with parameter set (n, k, 驴, μ), then, unless k is even and μ is odd, the binary hull is non-trivial, of minimum weight generally greater than 2k 驴 2, and we construct words of low weight in the hull; if k 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, k 驴 3, that has an ℓ-cycle for some ℓ 驴 3, the binary hull is shown to be non-trivial with minimum weight at most 2ℓ(k驴2). Properties of the p-ary hulls are also established.