List-decoding of binary Goppa codes up to the binary Johnson bound

Abstract: We study the list-decoding problem of alternant codes, with the notable case of classical Goppa codes. The major consideration here is to take into account the size of the alphabet, which shows great influence on the list-decoding radius. This amounts to compare the \emph{generic} Johnson bound to the \emph{$q$-ary} Johnson bound. This difference is important when $q$ is very small. Essentially, the most favourable case is $q=2$, for which the decoding radius is greatly improved, notably when the relative minimum distance gets close to 1/2. Even though the announced result, which is the list-decoding radius of binary Goppa codes, is new, it can be rather easily made up from previous sources (V. Guruswami, R. M. Roth and I. Tal, R .M. Roth), which may be a little bit unknown, and in which the case of binary Goppa codes has apparently not been thought at. Only D. J. Bernstein treats the case of binary Goppa codes in a preprint. References are given in the introduction. We propose an autonomous treatment and also a complexity analysis of the studied algorithm, which is quadratic in the blocklength $n$, when decoding at some distance of the relative maximum decoding radius, and in $O(n^7)$ when reaching the maximum radius.