Two families of Niho sequences having four-valued cross correlation with m-sequences

Abstract

For two odd integers $m$ and $s$ with $1\leq~s<m$ and ${\rm~gcd}(m,s)=1$, let $h$ satisfy $h(2^{s}-1)\equiv1~({\rm~mod}\,2^{m}+1)$ and $d=(h+1)(2^{m}-1)+1$.The cross correlation function between a binary $m$-sequence of period $2^{2m}-1$ and its $d$-decimation sequence is proved to take four values, and the correlation distribution is completely determined. Let $n$ be an even integer and $k$ be an integer with $1\leq~k\leq~\frac{n}{2}$. For an odd prime $p$ and a $p$-ary $m$-sequence $\{s(t)\}$ of period $p^{n}-1$, define $u(t)=\sum_{i=0}^{\frac{p^{k}-1}{2}}s(d_{i}t)$, where $d_{i}=ip^{\frac{n}{2}}+p^{k}-i$ and $i=0,1,\ldots,~\frac{p^{k}-1}{2}$. It is proved that the cross correlation function between $\{u(t)\}$ and $\{s(t)\}$ is three-valued or four-valued depending on whether $k$ is equal to$\frac{n}{2}$ or not, and the distribution is also determined.