  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.
