Almost supplementary difference sets and quaternary sequences

TL;DR

This paper introduces almost supplementary difference sets (ASDS) and demonstrates their equivalence to optimal autocorrelation quaternary sequences, providing new existence results and exploring their connections to binary sequences and group cohomology.

Contribution

The paper defines ASDS, establishes their link to optimal quaternary sequences, and proves existence under specific number-theoretic conditions.

Findings

ASDS are equivalent to quaternary sequences with optimal autocorrelation.

Existence of certain ASDS when 2m-1 is a prime power or m ≡ 1 mod 4 is prime.

Connections between ASDS, binary sequences, and group cohomology are explored.

Abstract

We introduce almost supplementary difference sets (ASDS). For odd $m$, certain ASDS in ${\mathbb Z}_m$ that have amicable incidence matrices are equivalent to quaternary sequences of odd length $m$ with optimal autocorrelation. As one consequence, if $2m-1$ is a prime power, or $m \equiv 1 \mod 4$ is prime, then ASDS of this kind exist. We also explore connections to optimal binary sequences and group cohomology.