Minimum distance bounds for cyclic codes and Deligne's theorem

Abstract

At the present time, there are very good methods to obtain bounds for the minimum distance of BCH codes and their duals. On the other hand, there are few other bounds suitable for general cyclic codes. Therefore, research Problem 9.9 of MacWilliams and Sloane (1977), The Theory of Error-Correcting Codes, asks if the bound of Deligne (1974) for exponential sums in several variables or the bound of Lang and Weil (1954), can be used to obtain bounds on the minimum distance of codes. This question is answered in the affirmative by showing how Deligne's theorem can be made to yield a lower bound on the minimum distance of certain classes of cyclic codes. In the process, an infinite family of binary cyclic codes is presented for which the bound on minimum distance so derived is as tight as possible. In addition, an infinite family of polynomials of degree 3 in 2 variables over a field of characteristic 2, for which Deligne's bound is tight, is exhibited. Finally, a bound is presented for the minimum distance of the duals of the binary subfield subcodes of generalized Reed-Muller codes as well as for the corresponding cyclic codes. It is noted that these codes contain examples of the best binary cyclic codes.