至多具有四值互相关的周期不同的$m$ -序列
Period-Different $m$-Sequences With At Most A Four-Valued Cross Correlation
摘要：
本文沿用Helleseth，Kholosha，Johanssen和Ness的最新工作，研究了周期为$2^m-1$的$m$ -序列与周期为较短的$2^{n}-1$的$m$ -序列的$d$ -抽取之间的互相关.假设$d$满足$d(2^l+1)=2^i({\rm mod} 2^n-1)$，对于某些$l$和$i$，我们证明了交叉相关恰好取三个或四个值，这取决于${\rm gcd}(l,n)$等于或大于1。相关值的分布也完全确定。我们的结果证实了Helleseth等人所发现的数值现象，并证明了除了本文所证明的情况外，没有其它$d$的情况能给出至多四值的互相关.
英文摘要：
In this paper, we follow the recent work of Helleseth, Kholosha, Johanssen and Ness to study the cross correlation between an $m$-sequence of period $2^m-1$ and the $d$-decimation of an $m$-sequence of shorter period $2^{n}-1$ for an even number $m=2n$. Assuming that $d$ satisfies $d(2^l+1)=2^i({\rm mod} 2^n-1)$ for some $l$ and $i$, we prove the cross correlation takes exactly either three or four values, depending on ${\rm gcd}(l,n)$ is equal to or larger than 1. The distribution of the correlation values is also completely determined. Our result confirms the numerical phenomenon Helleseth et al found. It is conjectured that there are no more other cases of $d$ that give at most a four-valued cross correlation apart from the ones proved here.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
