Zeta Functions for Self-Dual Codes

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The investigator and his colleagues study connections between self-dual linear codes and algebraic curves over a finite field. The main goal is to improve existing bounds for the asymptotic relative minimum distance of self-dual codes by showing that the zeros of the zeta function of a self-dual code satisfy a Riemann hypothesis analogue. The zeta function of an algebraic curve, that contains information about the number of rational points, can be defined for an arbitrary self-dual code. The zeta function of a self-dual code contains information about the minimum distance of the code, which has an expression in terms of the trace of the zeros of the zeta function. A Riemann hypothesis analogue would give an upper bound for the minimum distance similar to the Hasse-Weil bound for the number of rational points on an algebraic curve over a finite field. The problem under investigation comes from the theory of error-correcting codes. Error-correcting codes are used to ensure reliable communication over noisy communication channels. One of the central problems asks how good codes can be, in other words how many errors can be corrected if we choose an optimal code. Even if we restrict ourselves to the important and much smaller class of self-dual codes, no precise answer to this fundamental question is known, although some estimates are available. Improved estimates will follow by establishing analogues for self-dual codes to known results on zeta functions in Number Theory and Algebraic Geometry.