In this paper, for an integer k, we evaluate an upper bound for the cross-correlation of a ternary m-sequence of period N = 34k+2- 1 and its decimated sequence with decimation d = 34k+2- 32k+1+2/4 + 32k+1. It is found that the cross-correlation is upper bounded by 4.5 · 32k+1+ 1.

In this paper, for an odd prime , we investigate into the cross-correlation of a p-ary m-sequence m(t) of period p;n;-1 and its d-decimated sequences m(dt+l), 0≤l;m;+1)/2, where d=(pm+1)2/2(p+l), n=2m, and m is an odd integer. There are (pm+1)/2 distinct decimated sequences m(dt+l) since gcd(d,pn-1)=(pm+1)/2. It is shown that the magnitude of the cross-correlation values is upper bounded by (p+1)/2 pn/2+1 . We also construct the sequence family F from these sequences, where the family size is pmand the correlation magnitude is upper bounded by (pm+1)/2 pn/2+1.

For a prime p ≡ 1 mod 4, an odd integer n, and d = pn+1/p+1 + pn-1/2, we investigate the cross-correlation values of p-ary m-sequence m(t) of period pn- 1 and its decimated m-sequence m(dt). It is shown that the cross-correlation function between m(t) and m(dt) takes the values in {-1, -1 ± pn/2, -1 ± 1+√p/2 pn/2, -1 ± p-1/2 pn/2}.

In this paper, we derive the cyclotomic numbers of order 5 over an extension field Fpnusing the well-known results of quintic Jacobi sums over Fp(B. C. Berndt, et al., 1998). For p ne 1 mod 5, we have obtained the simple closed-form expression of the cyclotomic numbers of order 5 over Fpn. For p equiv 1 mod 5, we express the cyclotomic number of order 5 over Fpnin terms of the solution of the diophantine system which is required to evaluate the cyclotomic number of order 5 over Fpn. Using the cyclotomic numbers of order 5 over Fpn, autocorrelation distributions of 5-ary Sidel'nikov sequences of period pn- 1 are also derived