ALMOST PERFECT SEQUENCES

[PL, 1993] There is no $1$-APS of length $32$,$44$,$68$,$72$,$80$,$92$.

$s$ is a $\theta$-APS iff $\vert \hat s(u)\vert{}^2=\begin{cases}4\theta^2, &\text{$u$\space even;}\\2v-4\theta^2, &\text{$u$\space odd.}\end{cases}$

    Let $p$ be a prime. Let $m$ and $v$ be two integers. If ${\rm ord}_p(m)$ is an odd integer and if there exists $a$ s.t. $p^a=-1 \mod v$ then the equation has no solution in the ring ${\bf Z}[\zeta_v]$.

    Example $v=44$ and $m=2v-4=4.3.7$. Since $7^5 \equiv -1 \mod 44$...

    main conjecture

there exists a $1$-APS of length $v$ if and only if $\frac v2 -1$ is a prime power.