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

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 F2 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.