Hulls of codes from incidence matrices of connected regular graphs

The hulls of codes from the row span over , 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 , 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, lambda, mu), then, unless k is even and mu 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 mu is odd, we show that the binary hull is zero. Further, if a graph is the line graph of a k-regular graph, k a parts per thousand yen 3, that has an a""-cycle for some a"" a parts per thousand yen 3, the binary hull is shown to be non-trivial with minimum weight at most 2a""(k-2). Properties of the p-ary hulls are also established.