Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Interpolating polynomials from their values
Journal of Symbolic Computation - Special issue on computational algebraic complexity
A survey of partial difference sets
Designs, Codes and Cryptography
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
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
Preconditioners for singular black box matrices
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Efficient matrix rank computation with application to the study of strongly regular graphs
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Recent progress in algebraic design theory
Finite Fields and Their Applications
Quadratic-time certificates in linear algebra
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Fast computation of Smith forms of sparse matrices over local rings
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
| Hi-index | 0.00 |
For the problem of computing the rank of a matrix we have a complexity result and a practical implementation, both of which apply best to the case of a matrix whose rank is substantially smaller than its order. First, matrix rank can be computed in essentially optimal time when the rank is sufficiently small, specifically when the rank is less than the matrix order to the two-thirds power. Second, we have built software infrastructure to improve efficiency of matrix operations over a small finite field. Using this we have computed the 3-ranks of the first 7 matrices of the sequence D(3,k). These are adjacency matrices of strongly regular graphs defined by a construction of Dickson based on operations in a semifield of order 32k. In particular, D(3,7) is a 4,782,969 by 4,782,969 matrix and it's rank computation uses four days of CPU time. On the basis of the first 7 values, we conjecture that the sequence of ranks of the D(3,k) satisfies a linear recurrence of degree three.