### Citation:

Ghinelli , D , Key , J D & McDonough , T 2014 , ' Hulls of codes from incidence matrices of connected regular graphs ' Designs, Codes and Cryptography , vol 70 , no. 1 , pp. 35-54 . DOI: 10.1007/s10623-012-9635-0

### Abstract:

The hulls of codes from the row span over FpFp , 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 FpFp , where A is an adjacency matrix for the graph. Ifp = 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 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.

### Description:

D.Ghinelli, J.D.Key, T.P.McDonough. Hulls of codes from incidence matrices of connected regular graphs. Designs Codes and Cryptography, 2014, vol 70, pg 35-54.