A simple approach for construction of algebraic-geometric codes from affine plane curves

Abstract

The current algebraic-geometric (AG) codes are based on the theory of algebraic-geometric curves. In this paper we present a simple approach for the construction of AG codes, which does not require an extensive background in algebraic geometry. Given an affine plane irreducible curve and its set of all rational points, we can find a sequence of monomials x/sup i/y/sup j/ based on the equation of the curve. Using the first r monomials as a basis for the dual code of a linear code, the designed minimum distance d of the linear code, called the AG code, can be easily determined. For these codes, we show a fast decoding procedure with a complexity O(n/sup 7/3/), which can correct errors up to [(d-1/2]. For this approach it is neither necessary to know the genus of curve nor the basis of a differential form. This approach can be easily understood by most engineers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>