A family of sequences with large size and good correlation property arising from $M$-ary Sidelnikov sequences of period $q^d-1$

TL;DR

This paper constructs a large family of $M$-ary sequences with good correlation properties from Sidelnikov sequences, extending previous work for the case when $d=2$, and provides bounds on their correlation magnitude and size.

Contribution

It introduces a new family of sequences derived from Sidelnikov sequences for general $d$, with proven bounds on correlation and asymptotic size, extending prior results.

Findings

Maximum correlation magnitude bounded by $(2d -1) \\sqrt{q} + 1$

Asymptotic size of the sequence family is approximately $(M-1)q^{d-1}/d$

Extends previous work for the case $d=2$ to general $d$.

Abstract

Let $q$ be any prime power and let $d$ be a positive integer greater than 1. In this paper, we construct a family of $M$-ary sequences of period $q-1$ from a given $M$-ary, with $M|q-1$, Sidelikov sequence of period $q^d-1$. Under mild restrictions on $d$, we show that the maximum correlation magnitude of the family is upper bounded by $(2d -1) \sqrt { q }+1$ and the asymptotic size, as $q\rightarrow \infty$, of that is $\frac{ (M-1)q^{d-1}}{d }$. This extends the pioneering work of Yu and Gong for $d=2$ case.