A Novel 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 novel approach for construction of AG codes without any background in algebraic geometry. Given an affine plane irreducible curve and its all rational points, based on the equation of this curve, we can find a sequence of monomial polynomials x/sup i/y/sup j/. Using the first r polynomials as a basis of dual code of a linear code called AG code, the designed minimum distance d of this AG code can be easily determined. For these codes a fast decoding procedure with complexity O(n/sup 7/3/) which can correct errors up to [(d-1)/2], is also shown. By this approach it is neither necessary to know the genus of curve nor find a basis of differential form. This approach can be easily understood by most engineers. Some examples are also shown, which indicate that the codes constructed by this approach are better than the current AG codes from same curves.