On the Structure of Cyclotomic Fourier Transforms and Their Applications to Reed-Solomon Codes

Abstract

This paper is focused on cyclotomic Fourier transforms in GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ), and on their applications to algebraic decoding of Reed-Solomon codes, like the evaluation of syndromes and of error locator (or evaluator) polynomials. Cyclotomic transforms are much more efficient than straightforward evaluation. In particular, the number of multiplications is quite small. In this paper it is shown that also the number of additions can be considerably reduced with respect to previous analyses. A simple interpretation of the cyclotomic Fourier transform best suited for the evaluation of syndromes allows to assemble the required matrix easily and quickly, even in large fields. Fast construction of such matrices is important to obtain the best results, since as many matrices as possible must be generated and compared. It is shown that both the structure of the matrix and of bilinear convolutions need to be exploited, to reduce the complexity of the costly part of cyclotomic Fourier transforms, which is a matrix-vector product. Heuristic algorithms for matrix-vector product are to be run as many times as possible to obtain the best transform. It is shown with several examples that very good results can be obtained even with very simple algorithms.