Algebro-geometric codes on curves of small genera. (English. Russian original) Zbl 0636.94014

Barg, A. M.; Katsman, G. L.; Tsfasman, M. A.

Probl. Inf. Transm. 23, No. 1-2, 34-38 (1987); translation from Probl. Peredachi Inf. 23, No. 1, 42-46 (1987).

In a by now famous paper, M. A. Tsfasman, S. G. Vlǎdut and Th. Zink [Math. Nachr. 109, 21–28 (1982; Zbl 0574.94013)] showed a construction of very good codes using methods from algebraic geometry and a number of ideas essentially due to V. D. Goppa. The paper led to a series of papers and constructions of special examples. In the present paper a variant of Goppa’s construction due to Manin [S. G. Vlǎdut and Yu. I. Manin, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 25, 209–257 (1984; Zbl 0629.94013)] is first presented. The curves that are considered have genus \(1, 2\), or \(3\). The main results are constructions of several codes that are better than the best presently known linear codes by using the geometric code as an outer code and then using the nested-concatenation ideas of È. L. Blokh and V. V. Zyablov [Linear concatenated codes (Russian) Linear concatenated codes. (Russian) Moskva: Nauka (1982)] for the choice of inner codes. Even very simple concatenations such as replacing elements of \(\mathbb F_8\) by columns from \(\mathbb F_4\), using an overall parity check, sometimes already lead to improvements.

Reviewer: J. H. van Lint (Eindhoven)

MSC:

94B27	Geometric methods (including applications of algebraic geometry) applied to coding theory
14G50	Applications to coding theory and cryptography of arithmetic geometry

Keywords:

algebro-geometric codes

Citations:

Zbl 0574.94013; Zbl 0629.94013