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

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Güneri, Cem and Özbudak, Ferruh (2008) Multidimensional cyclic codes and Artin–Schreier type hypersurfaces over finite fields. Finite Fields and Their Applications, 14 (1). pp. 44-58. ISSN 1071-5797

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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 a±ne rational points of Artin-Schreier hypersurfaces defined over certain finite fields. Using Deligne’s and Hasse- Weil-Serre inequalities we state bounds on the minimum distance. Comparison of the bounds is made and illustrated by examples. Some applications of our results are given. Over F2, we obtain a bound on certain character sums giving better estimates than Deligne’s inequality in some cases. We improve the minimum distance bounds of Moreno-Kumar on p-ary subfield subcodes of generalized Reed-Muller codes for some parameters. We also characterize qm- optimal and maximal Artin-Schreier hypersurfaces.
Item Type: Article
Uncontrolled Keywords: Multidimensional cyclic code; Artin-Schreier hypersurface; Deligne’s inequality; Hasse-Weil-Serre inequality.
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Engineering and Natural Sciences
Depositing User: Cem Güneri
Date Deposited: 26 Oct 2007 16:03
Last Modified: 18 Feb 2014 11:16
URI: https://research.sabanciuniv.edu/id/eprint/5907

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