A transform approach to Goppa codes

Based on a finite Fourier transform of codewords over the finite field GF (q^{m}) , 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.

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