In this paper, for an odd prime p, two positive integers n, m with n=2m, and pm≡1 (mod 4), we derive an upper bound on the magnitude of the cross-correlation function between two decimated sequences of a p-ary m-sequence. The two decimation factors are 2 and 2(pm+1), and the upper bound is derived as $rac{3}{2}p^m + rac{1}{2}$. In fact, those two sequences correspond to the p-ary sequences used for the construction of Kasami sequences decimated by 2. This result is also used to obtain an upper bound on the cross-correlation magnitude between a p-ary m-sequence and its decimated sequence with the decimation factor $d=rac{(p^m +1)^2}{2}$.

In this paper, for an odd prime p and i=0,1, we investigate the cross-correlation between two decimated sequences, s(2t+i) and s(dt), where s(t) is a p-ary m-sequence of period pn-1. Here we consider two cases of ${d}$, ${d=rac{(p^m +1)^2}{2} }$ with ${n=2m}$, ${p^m equiv 1 pmod{4}}$ and ${d=rac{(p^m +1)^2}{p^e + 1}}$ with n=2m and odd m/e. The value distribution of the cross-correlation function for each case is completely determined. Also, by using these decimated sequences, two new p-ary sequence families of period ${rac{p^n -1}{2}}$ with good correlation property are constructed.

Let p be an odd prime and m be any positive integer. Assume that n=2m and e is a positive divisor of m with m/e being odd. For the decimation factor $d=rac{(p^{m}+1)^2}{2(p^e+1)}$, the cross-correlation between the p-ary m-sequence ${tr_1^n(alpha^i)}$ and its decimated sequence ${tr_1^n(alpha^{di})}$ is investigated. The value distribution of the correlation function is completely determined. The results in this paper generalize the previous results given by Choi, Luo and Sun et al., where they considered some special cases of the decimation factor d with a restriction that m is odd. Note that the integer m in this paper can be even or odd. Thus, the decimation factor d here is more flexible than the previous ones. Moreover, our method for determining the value distribution of the correlation function is different from those adopted by Luo and Sun et al. in that we do not need to calculate the third power sum of the correlation function, which is usually difficult to handle.

Let m ≥ 3 be an odd positive integer, n=2m and p be an odd prime. For the decimation factor $d=rac{(p^{m}+1)(p^{m}+p-1)}{p+1}$, the cross-correlation between the p-ary m-sequence {tr1n(αt)} and its all decimated sequences {tr1n(αdt+l)} is investigated, where 0 ≤ l < gcd(d,pn-1) and α is a primitive element of Fpn. It is shown that the cross-correlation function takes values in {-1,-1±ipm|i=1,2,…p}. The result presented in this paper settles a conjecture proposed by Kim et al. in the 2012 IEEE International Symposium on Information Theory Proceedings paper (pp.1014-1018), and also improves their result.

In this paper, for an odd prime p such that p≡3 mod 4, odd n, and d=(pn+1)/(pk+1)+(pn-1)/2 with k|n, the value distribution of the exponential sum S(a,b) is calculated as a and b run through $mathbb{F}_{p^n}$. The sequence family $mathcal{G}$ in which each sequence has the period of N=pn-1 is also constructed. The family size of $mathcal{G}$ is pn and the correlation magnitude is roughly upper bounded by $(p^k+1)sqrt{N}/2$. The weight distribution of the relevant cyclic code C over $mathbb{F}_p$ with the length N and the dimension ${ m dim}_{mathbb{F}_p}mathcal{C}=2n$ is also derived.

In this paper, we analyze the existing results to derive the cross-correlation distributions of p-ary m-sequences and their decimated sequences for an odd prime p and various decimations d. Based on the previously known results, a methodology to obtain the distribution of their cross-correlation values is also formulated.