Improved p-ary Codes and Sequence Families from Galois Rings

This paper explores the applications of a recent bound on some Weil-type exponential sums over Galois rings in the construction of codes and sequences. A family of codes over Fp, mostly nonlinear, of length pm+1 and size p2 ·pm(D− D/p ), where 1 ≤ D ≤ pm/2, is obtained. The bound on this type of exponential sums provides a lower bound for the minimum distance of these codes. Several families of pairwise cyclically distinct p-ary sequences of period p(pm − 1) of low correlation are also constructed. They compare favorably with certain known p-ary sequences of period pm − 1. Even in the case p = 2, one of these families is slightly larger than the family Q(D) in section 8.8 in [T. Helleseth and P. V. Kumar, Handbook of Coding Theory, Vol. 2, North-Holland, 1998, pp. 1765–1853], while they share the same period and the same bound for the maximum nontrivial correlation.