Sequences with thirteen-valued cross correlations

In this paper, we completely determine the cross correlation distribution between an $m$-sequence $(s_t)$ of period $p^n-1$ and its $d$-decimated sequence $(s_{dt})$, where $d=\frac{p^n-1}{3}+p^i$, $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$, $\frac{1}{3}{p}^{-i}(p^n-1)\equiv ̸2(\mathrm{m}\mathrm{o}\mathrm{d}3)$, and $0\le i<n$. It is shown that the cross correlation is $13$-valued. To the best of our knowledge, this is the first time that the cross correlation distribution of so many values has been determined.