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

We obtain a trace representation for multidimensional cyclic codes via Delsarteʹs theorem. This relates the weights of the codewords to the number of affine rational points of Artin–Schreier type hypersurfaces over finite fields. Using Deligneʹs 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 which gives better estimates than Deligneʹs 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.