Multidimensional cyclic codes and Artin–Schreier hypersurfaces over finite fields

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.

Available Versions of this Item

• Multidimensional cyclic codes and Artin–Schreier hypersurfaces over finite fields. (deposited 26 Oct 2007 14:21) [Currently Displayed]
  • Multidimensional cyclic codes and Artin–Schreier type hypersurfaces over finite fields. (deposited 26 Oct 2007 16:03)