Improved p-ary Codes and Sequence Families from Galois Rings of Characteristic p2

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 $\F_p$, mostly nonlinear, of length $p^{m+1}$ and size $p^2 \cdot p^{m ( D - \lfloor D/p^2 \rfloor )}$, where $1 \le D \le p^{m/2}$, 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 $p(p^m-1)$ 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 $Q(D)$ 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.