On the Cross-Correlation Distribution of p-ary m-Sequences

For an odd prime p, an even integer n, and d = p + 1 with gcd(n, k) = 1, there are p + 1 distinct decimated sequences s(dt + l), 0 ≤ l < p + 1, for a p-ary m-sequence s(t) of period p − 1 since gcd(d, p − 1) = p + 1. In this paper, the cross-correlation distribution between a p-ary m-sequence s(t) and its p+1 distinct decimated sequences s(dt+ l) is derived. The maximum magnitude Cmax of their cross-correlation values is 1+p √ pn when l = 0 mod p+1 for n = 0 mod 4 or when l = (p+ 1)/2 mod p+1 for n = 2 mod 4. For the remaining cases, Cmax is 1+ √ pn. Also by using s(t) and s(dt+ l), a new family of p-ary sequences of period p − 1 is constructed, whose family size is p and Cmax is 1 + p √ pn.