Let p be an odd prime and n = 2m with pm≡ 1 (mod 4). In this paper, the cross-correlation distribution between two decimated sequences of a p-ary m-sequence, s(2t + i) with i ∈ {0, 1} and s(d't) with d' = 2d, d = (pm+1/2)2is determined.

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.

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 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.