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 F-2 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. (c) 2006 Elsevier Inc. All rights reserved.