Hulls of codes from incidence matrices of connected regular graphs

The hulls of codes from the row span over Fp, 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 Fp, 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 in [8] 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.