Fast Solutions of Toeplitz Equations over Finite Fields

Abstract

Fast algorithms for solving Toeplitz systems of equations, such as the Levinson algorithm, are well known. However, in practice the submatrices occurring the Levinson algorithm may be poorly conditioned, even if the overall Toeplitz matrix is well conditioned. The problem is particularly severe when the equations are over a finite field: many of the submatrices may be singular. Approaches to this problem developed over floating point numbers using a “look-ahead” Levinson algorithm, do not adequately address the problem over finite fields. Singular computations can still arise in the situation when there are multiple successive look-ahead steps. This paper describes how the problem arises and provides a solution to the finite-field solution that works in all cases. In the case of multiple look-ahead steps, a recursive filling procedure is invoked at earlier steps in the algorithm in some instances. A discussion of the distribution of the block sizes in the look-ahead algorithm is also provided.