Shift-Inequivalent Decimations of the Sidelnikov-Lempel-Cohn-Eastman Sequences

TL;DR

This paper investigates the shift-inequivalent decimations of SLCE sequences, establishing conditions for their multipliers and showing that for odd primes, the multiplier group is trivial, leading to a large family of sequences with good autocorrelation.

Contribution

It provides a necessary numerical condition for multipliers of SLCE almost difference sets and proves the triviality of the multiplier group for odd primes, enabling construction of many inequivalent sequences.

Findings

Multiplier group of SLCE sets over odd primes is trivial.

Constructs a family of (p-1) shift-inequivalent sequences with good autocorrelation.

Provides a necessary condition for residues to be multipliers of SLCE sets.

Abstract

We consider the problem of finding maximal sets of shift-inequivalent decimations of Sidelnikov-Lempel-Cohn-Eastman (SLCE) sequences (as well as the equivalent problem of determining the multiplier groups of the almost difference sets associated with these sequences). We derive a numerical necessary condition for a residue to be a multiplier of an SLCE almost difference set. Using our necessary condition, we show that if $p$ is an odd prime and $S$ is an SLCE almost difference set over $\mathbb{F}_p,$ then the multiplier group of $S$ is trivial. Consequently, for each odd prime $p,$ we obtain a family of $\phi(p-1)$ shift-inequivalent balanced periodic sequences (where $\phi$ is the Euler-Totient function) each having period $p-1$ and nearly perfect autocorrelation.