Designs, Codes and Cryptography

, Volume 70, Issue 1–2, pp 35–54 | Cite as

Hulls of codes from incidence matrices of connected regular graphs

Article

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−2. Here we show that if the graph is strongly regular with parameter set (n, k, λ, 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 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 ≥ 3, that has an -cycle for some ≥ 3, the binary hull is shown to be non-trivial with minimum weight at most 2(k−2). Properties of the p-ary hulls are also established.

Keywords

Incidence matrix Graph Code Hull Permutation decoding 

Mathematics Subject Classification

05B05 05C38 94B05 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma ‘La Sapienza’RomeItaly
  2. 2.Department of Mathematics and Applied MathematicsUniversity of the Western CapeBellvilleSouth Africa
  3. 3.Institute of Mathematics and PhysicsAberystwyth UniversityAberystwythUK

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