List Decoding of q - ary Reed - Muller Codes 1 )

The q-ary Reed-Muller codes RMq(u,m) of length n = q are a generalization of Reed-Solomon codes, which use polynomials in m variables to encode messages through functional encoding. Using an idea of reducing the multivariate case to the univariate case, randomized list-decoding algorithms for Reed-Muller codes were given in [1] and [15]. The algorithm in [15] is an improvement of the algorithm in [1], it is applicable to codes RMq(u,m) with u < q/2 and works for up to E < n(1 − √ 2u/q) errors. In this paper, following [6], we show that q-ary Reed-Muller codes are subfield subcodes of Reed-Solomon codes over Fqm . Then, using the list-decoding algorithm in [5] for ReedSolomon codes over Fqm , we present a list-decoding algorithm for q-ary Reed-Muller codes. This algorithm is applicable to codes of any rates, and achieves an error-correction bound n(1− √ (n− d)/n). The algorithm achieves a better error-correction bound than the algorithm in [15], since when u is small, n(1 − √ (n− d)/n) = n(1 − √ u/q). The implementation of the algorithm requires O(n) field operations in Fq and O(n) field operations in Fqm under some minor assumption.