A remark on algebraic geometric codes. (English) Zbl 0674.94012

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

Representation theory, group rings, and coding theory, Pap. in Honor of S. D. Berman, Contemp. Math. 93, 197-199 (1989).

Summary: [For the entire collection see Zbl 0666.00005.]
In [Izv. Akad. Nauk SSSR, Ser. Mat. 46, 762-781 (1982; Zbl 0522.94013)] V. D. Goppa mentioned that for algebraic geometric codes restricted to a subfield the parameters sometimes can be in fact better than the trivial estimate (just as for BCH-codes). We prove a respective theorem, and apply it to the study of polynomial asymptotic bounds. The bound established in [S. G. Vlehduts, G. L. Katsman and M. A. Tsfasman, Probl. Peredachi Inf. 20, No.1, 47-55 (1984; Zbl 0555.94008)] is ameliorated so that for \(\delta\) \(\to 0\) the asymptotic behaviour of the obtained bound coincides (for \(q=2)\) with that of the Gilbert- Varshamov bound.

MSC:

94B05

Linear codes, general

Keywords:

algebraic geometric codes; Gilbert-Varshamov bound