The cross-correlation distribution of a pp-ary mm-sequence of period p2m−1p^{2m}-1 and its decimation by (pm+1)22(pe+1)\frac{(p^{m}+1)^{2}}{2(p^{e}+1)}

Let $n=2m$, $m$ odd, $e|m$, and $p$ odd prime with $p\equiv1\ \mathrm{mod}\ 4$. Let$d=\frac{(p^{m}+1)^{2}}{2(p^{e}+1)}$. In this paper, we study the cross-correlation between a$p$-ary$m$-sequence$\{s_{t}\}$of period$p^{2m}-1$and its decimation$\{s_{dt}\}$. Our result shows that the cross-correlation function is six-valued and that it takes the values in$\{-1,\ \pm p^{m}-1,\ \frac{1\pm p^{\frac{e}{2}}}{2}p^{m}-1,\ \frac{(1- p^{e})}{2}p^{m}-1\}$. Also, the distribution of the cross-correlation is completely determined