A "coordinate recurrence" method for solving sparse systems of linear equations over finite fields is described. The algorithms discussed all requireO(n_{1}(\omega + n_{1})\log^{k}n_{1})field operations, wheren_{1}is the maximum dimension of the coefficient matrix,\omegais approximately the number of field operations required to apply the matrix to a test vector, and the value ofkdepends on the algorithm. A probabilistic algorithm is shown to exist for finding the determinant of a square matrix. Also, probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error.

Two interesting cosets of the first-order Reed-Muller code of block length2^{15}are described. They provide counterexamples to a conjecture on the covering radius.