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

We study the list-decoding problem of alternant codes (which includes obviously that of classical Goppa codes). The major consideration here is to take into account the (small) size of the alphabet. This amounts to comparing the generic Johnson bound to the q-ary Johnson bound. The most favourable case is q = 2, for which the decoding radius is greatly improved. Even though the announced result, which is the list-decoding radius of binary Goppa codes, is new, we acknowledge that it can be made up from separate previous sources, which may be a little bit unknown, and where the binary Goppa codes has apparently not been thought at. Only D. J. Bernstein has treated the case of binary Goppa codes in a preprint. References are given in the introduction. We propose an autonomous and simplified treatment and also a complexity analysis of the studied algorithm, which is quadratic in the blocklength n, when decoding e-away of the relative maximum decoding radius.

I. Introduction

In 1997, Sudan presented the first list-decoding algorithm for Reed-Solomon codes [1]. Since the correction radius of Sudan's algorithm is larger, this represented an important milestone in algorithmic list-decoding Afterwards, Guruswami and Sudan [2] improved the previous algorithm by adding the notion of multiplicities. The number of errors that this algorithm is able to list-decode corresponds to the Johnson radius , where is the minimum distance of the code.