Efficient matrix rank computation with application to the study of strongly regular graphs

We present algorithms for computing the p-rank of integer matrices. They are designed to be particularly effective when p is a small prime, the rank is relatively low, and the matrix itself is large and dense and may exceed virtual memory space. Our motivation comes from the study of difference sets and partial difference sets in algebraic design theory. The p-rank of the adjacency matrix of an associated strongly regular graph is a key tool for distinguishing difference set constructions and thus answering various existence questions and conjectures. For the p-rank computation, we review several memory efficient methods, and present refinements suitable to the small prime, small rank case. We give a new heuristic approach that is notably effective in practice as applied to the strongly regular graph adjacency matrices. It involves projection to a matrix of order slightly above the rank. The projection is extremely sparse, is chosen according to one of several heuristics, and is combined with a small dense certifying component. Our algorithms and heuristics are implemented in the LinBox library. We also briefly discuss some of the software design issues and we present results of experiments for the Paley and Dickson sequences of strongly regular graphs.