A transform approach to Goppa codes

Abstract

Based on a finite Fourier transform of codewords over the finite field GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q^{m})</tex> , some basic properties of the class of Goppa codes are presented. A new lower bound on the minimum distance for these codes is derived and applied to a subclass of Goppa codes which is. in fact. equivalent to a subset of punctured reversible cyclic codes. Furthermore. it is shown how the class of Goppa codes can be easily decoded in the context of this transformation by using the Berlekamp-Massey decoding algorithm. Through a slight extension of the procedure, it is also shown how this algorithm may be used m decode alternant codes up to their guaranteed error-correction capability.