Construction of orientable sequences in $O(1)$-amortized time per bit

Abstract

An orientable sequence of order $n$ is a cyclic binary sequence such that each length-$n$ substring appears at most once \emph{in either direction}. Maximal length orientable sequences are known only for $n\le 7$, and a trivial upper bound on their length is $2^{n-1}-2^{\lfloor (n-1)/2\rfloor }$. This paper presents the first efficient algorithm to construct orientable sequences that reach this upper bound, asymptotically; more specifically, our algorithm constructs orientable sequences via cycle-joining and a successor-rule approach requiring $O(n)$ time per bit and $O(n)$ space. This answers a longstanding open question from Dai, Martin, Robshaw, Wild [Cryptography and Coding III (1993)]. Applying a recent concatenation-tree framework, the same sequences can be generated in $O(1)$-amortized time per bit using $O(n^2)$ space. Our sequences are applied to find new longest-known (aperiodic) orientable sequences for $n\le 20$.