In this article we see quasi-cyclic codes as block cyclic codes. We generalize some properties of cyclic codes to quasi-cyclic codes. We show a one-to-one correspondence between ℓ-quasi-cyclic codes of length mℓ and left ideals of M ℓ (Fq)[X]/(X m − 1). Then, we generalize BCH codes and evaluation codes in this context. We study their parameters and establish a key equation. Finally, we present a new [189,11, 125] F 4 code beating the known minimum distance for fixed length and dimension. Many codes with good parameters beating best known ones have been found from this latter.
Abstract-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 -away of the relative maximum decoding radius.
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