Simplified high-speed high-distance list decoding for alternant codes

Abstract

This paper presents a simplified list-decoding algorithm to correct any number w of errors in any alternant code of any length n with any designed distance t + 1 over any finite field Fq; in particular, in the classical Goppa codes used in the McEliece and Niederreiter public-key cryptosystems. The algorithm is efficient for w close to, and in many cases slightly beyond, the Fq Johnson bound where J′ = n′ - √ n′(n′ - t - 1) where n′ = n(q - 1)/q, assuming t + 1 ≤ n′. In the typical case that qn/t ε (lg n)O(1) and that the parent field has (lg n)O(1) bits, the algorithm uses n(lg n)O(1) bit operations for w ≤ J′ - n/(lg n)O(1); O(n4.5) bit operations for w ≤ J′ + o((lg n)/lg lg n); and nO(1) bit operations for w ≤ J′ + O((lg n)/lg lg n). © 2011 Springer-Verlag.