On the Cross-Correlation between Two Decimated p-Ary m-Sequences by 2 and 4pn/2-2

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, $\bigg \{\frac{-1 \pm p^{n/2}}{2}, \frac{-1 + 3p^{n/2}}{2}, \frac{-1 + 5p^{n/2}}{2} \bigg \}$. From this result, for pm ≠ 2 mod 3, a new sequence family with family size 4N and the maximum correlation magnitude upper bounded by $\frac{-1 + 5p^{n/2}}{2} \simeq \frac{5}{\sqrt{2}}\sqrt{N}$ is constructed, where $N = \frac{p^n-1}{2}$ is the period of sequences in the family.