A SIMPLE APPROACH FOR CONSTRUCTION OF ALGEBRAIC-GEOMETRIC CODES FROM AFFINE PLANE-CURVES

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 c urve and its set of all rational points, we can find a sequence of mon omials x(i)y(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 easi ly determined. For these codes, we show a fast decoding procedure with a complexity O(n7/3), which can correct errors up to [(d-1)/2]. For t his approach it is neither necessary to know the genus of curve nor th e basis of a differential form. This approach can be easily understood by most engineers.