Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
On the p-Rank of the Adjacency Matrices of Strongly Regular Graphs
Journal of Algebraic Combinatorics: An International Journal
A survey of partial difference sets
Designs, Codes and Cryptography
Certifying inconsistency of sparse linear systems
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
Integer Smith form via the valence: experience with large sparse matrices from homology
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
On Wiedemann's Method of Solving Sparse Linear Systems
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Solving Large Sparse Linear Systems over Finite Fields
CRYPTO '90 Proceedings of the 10th Annual International Cryptology Conference on Advances in Cryptology
Finite field linear algebra subroutines
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
An output-sensitive variant of the baby steps/giant steps determinant algorithm
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
FFPACK: finite field linear algebra package
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Preconditioners for singular black box matrices
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
A block Wiedemann rank algorithm
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
A family of skew Hadamard difference sets
Journal of Combinatorial Theory Series A
Recent progress in algebraic design theory
Finite Fields and Their Applications
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Simultaneous modular reduction and Kronecker substitution for small finite fields
Journal of Symbolic Computation
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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.