Sequences with thirteen-valued cross correlations

TL;DR

This paper determines the complete cross correlation distribution between an m-sequence and its decimated version, revealing a 13-valued distribution for the first time in such sequences.

Contribution

It provides the first explicit determination of the 13-valued cross correlation distribution for specific m-sequences and their decimated sequences.

Findings

Cross correlation is 13-valued.

Complete distribution is explicitly determined.

First such determination for sequences with this many correlation values.

Abstract

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 $d = \frac{p^n-1}{3} + p^i$, $p \equiv 1 \pmod{3}$, $\frac{1}{3}p^{-i}(p^n-1) \not\equiv 2 \pmod{3}$, and $0 \leq i < n$. 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.