Hulls of codes from incidence matrices of connected regular graphs

The hulls of codes from the row span over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{F}_p}$$\end{document} , 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{F}_p}$$\end{document} , 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 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.