Almost supplementary difference sets and quaternary sequences

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 $m\equiv 1\mathrm{mod}4$ is prime, then ASDS of this kind exist. We also explore connections to optimal binary sequences and group cohomology.