Multidimensional cyclic codes and Artin-Schreier type hypersurfaces over finite fields

Guneri C., Ozbudak F.

FINITE FIELDS AND THEIR APPLICATIONS, vol.14, no.1, pp.44-58, 2008 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 14 Issue: 1
  • Publication Date: 2008
  • Doi Number: 10.1016/j.ffa.2006.12.003
  • Journal Name: FINITE FIELDS AND THEIR APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.44-58
  • Keywords: multidimensional cyclic code, Artin-Schreier type hypersurface, Deligne's inequality, Hasse-Weil-Serre inequality
  • Middle East Technical University Affiliated: Yes

Abstract

We obtain a trace representation for multidimensional cyclic codes via Delsarte's theorem. This relates the weights of the codewords to the number of affine rational points of Artin-Schreier type hypersurfaces over finite fields. Using Deligne's and Hasse-Weil-Serre inequalities we get bounds on the minimum distance. Comparison of the bounds is made and illustrated by examples. Some applications of our results are given. We obtain a bound on certain character sums over F-2 which gives better estimates than Deligne's inequality in some cases. We also improve the minimum distance bounds of Moreno-Kumar on p-ary subfield subcodes of generalized Reed-Muller codes for some parameters. (c) 2006 Elsevier Inc. All rights reserved.