On a Fast Decoding Algorithm for Goppa Codes Defined on Certain Algebraic Curves with at Most One Higher Order Cusp

Abstract

We propose a fast decoding algorithm for a class of geometric Goppa codes defined on certain algebraic plane curves, associated with Artin-Schreier extensions of F/sub q/(x), introduced by Stichtenoth [3]. Although we do not attempt here to treat all the class of curves introduced by Stichtenoth, we do include certain elliptic, hyperelliptic and Hermitian curves. These curves are defined by the homogeneous equation Y/sup a/Z/sup b-a/ + YZ/sup b-1/ = X/sup b/ over an arbitrary finite field F/sub q/ of characteristic p, where a and b are relatively prime integers such that a = P/sup v/(V /spl epsiv/ N/sup */), a < b and the zeros of Y/sup a/ + y form an additive subgroup of F/sub q/ of order p/sup v/. The main step of the proposed algorithm is to solve a key equation studied by Porter, Shen and Pellikaan [1]. For this purpose, we derive explicit formulas for certain differential forms, which are used to construct the syndrome of the codes defined on the above-mentioned curves, and propose a modified version of Sakata's algorithm [4]. Further, we prove, in work inspired by Shen's study [2], that the Porter-Shen-Pellikaan key equation for codes defied on the curves treated here can be solved by using our modified Sakata algorithm with complexity O(d/sup 2//sub des/a + g/sup 2/a), where d/sub des/ is the designed minimum distance and g is the genus of the curve. The proposed decoding algorithm may be regarded as an extension of Shen's algorithm [2] for Hermitian codes to a wider class of codes. For certain hyperelliptic codes, this algorithm can decode up to [(d/sub des/ - 1)/2] errors with complexity O(n/sup 2/), where n is the word length of the code.