In this paper, for an odd prime p such that p ≡ 3 mod 4, odd n, and d = (pn+ 1)/(pk+ 1) + (pn- l)/2 with k|n, the value distribution of the exponential sum S(a, b) when a and b run through Fpn is calculated. The weight distribution of the relevant cyclic code C over Fpwith the length L = pn- 1 and the dimension dimFpC = 2n In is also derived. Our result generalizes the case in [5].

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}.

For an odd primepcongruent to 3 modulo 4 and an odd integern, a new family ofp-ary sequences of periodN=(pn-1)/2 with low correlation is proposed. The family is constructed by shifts and additions of two decimated m-sequences with the decimation factors 2 and 2d,d=N-pn-1. The upper bound for the maximum magnitude of nontrivial correlations of this family is derived using well known Kloosterman sums. The upper bound is shown to be 2√(N+1/2) = √(2pn) , which is twice the Welch's lower bound and approximately 1.5 times the Sidelnikov's lower bound. The size of the family is 2(pn-1) , which is four times the period of sequences.

In this paper, we investigate into the cross-correlation of a ternary m-sequence m(t) of period 3n- 1 and its decimated sequence m(dt) by d = (3m+1)2/8, where n = 2m = 4k + 2. It is shown that the magnitude of the cross-correlation values is upper bounded by 2√(3n+ 1).

In this paper, we propose a method to construct p-ary sequence family S which has the period of N = (pn- 1)/2 when p is an odd prime and n is an odd integer. The family has the size of 2(pn- 1) and the magnitudes of correlation values between any two sequences from the family are upper bounded by 2√N + 1/2. The bound can be derived from the well-known Kloosterman sum.

In this paper, for a positive integer M and a prime p such that M|pn- 1, a family of M-ary sequences using the M-ary Sidel'nikov sequences with period pn-1 is constructed. This family has its maximum magnitude of correlation values upper bounded by 3radic(pn) + 6 and the family size is (M-I)2(2n-1-1) + M - 1 for p = 2 or (M-1)2(pn-3)/2+ M (M-1)/2 for an odd prime p.