Improved p-ary Codes and Sequence Families from Galois Rings

Abstract

In this paper, a recent bound on some Weil-type exponential sums over Galois rings is used in the construction of codes and sequences. The bound on these type of exponential sums provides a lower bound for the minimum distance of a family of codes over \({\mathbb F}_{p}\), mostly nonlinear, of length p m + 1 and size \(p^{2} \cdot p^{m(D-\lfloor \frac {D}{p^{2}}\rceil)}\), where 1 ≤ D ≤ p m/2. Several families of pairwise cyclically distinct p-ary sequences of period p(p m – 1) of low correlation are also constructed. They compare favorably with certain known p-ary sequences of period p m – 1. Even in the case p = 2, one of these families is slightly larger than the family Q(D) of [H-K, Section 8.8], while they share the same period and the same bound for the maximum non-trivial correlation.