Based on the work by Helleseth [1], for an odd prime p and an even integer n=2m, the cross-correlation values between two decimated m-sequences by the decimation factors 2 and 4pn/2-2 are derived. Their cross-correlation function is at most 4-valued, that is, $igg {rac{-1 pm p^{n/2}}{2}, rac{-1 + 3p^{n/2}}{2}, rac{-1 + 5p^{n/2}}{2} igg }$. From this result, for pm ≠ 2 mod 3, a new sequence family with family size 4N and the maximum correlation magnitude upper bounded by $rac{-1 + 5p^{n/2}}{2} simeq rac{5}{sqrt{2}}sqrt{N}$ is constructed, where $N = rac{p^n-1}{2}$ is the period of sequences in the family.

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.