Multidimensional cyclic codes and Artin–Schreier type hypersurfaces over finite fields
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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Official URL: http://dx.doi.org/10.1016/j.ffa.2006.12.003
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.
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