The cross-correlation distribution of a $p$-ary $m$-sequence of period $p^{2m}-1$ and its decimation by $\frac{(p^m+1{)}^2}{2(p^e+1)}$

Abstract

Let $n=2m$, $m$ odd, $e|m$, and $p$ odd prime with $p\equiv 1\mathrm{mod}4$. 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.