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. For an odd prime p, an even integer n, and d = pk +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 pn -1 because gcd(d, pn - 1) = p + 1. The maximum magnitude of their cross-correlation values is 1 + p if l ≡ 0 mod p + 1 for n ≡ 0 mod 4 or l ≡ (p + 1)/2 mod p + 1 for n ≡ 2 mod 4 and otherwise, 1 + . Also by using s(t) and s(dt + l), a new family of p-ary sequences of period pn -1 is constructed, whose family size is pn and Cmax is 1 + p.