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

TL;DR

This paper introduces an efficient algorithm for constructing long orientable binary sequences with optimal length, using cycle-joining and successor-rule methods, and achieves $O(1)$-amortized time per bit.

Contribution

It presents the first efficient construction method for asymptotically optimal orientable sequences, solving a longstanding open problem.

Findings

Sequences constructed for n ≤ 20 with new longest-known lengths.

Algorithm operates in O(n) time per bit and O(n) space.

Sequences can be generated in O(1) amortized time per bit with O(n^2) space.

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 $n\leq 7$, 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 with asymptotically optimal length; 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 $n\leq 20$.