Algebro-geometric codes on curves of small genera
In a by now famous paper, \textit{M. A. Tsfasman}, \textit{S. G. Vlǎdut} and \textit{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 [\textit{S. G. Vlǎdut} and \textit{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 \textit{È. L. Blokh} and \textit{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.
scientific article; zbMATH DE number 4037763