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 symbol and $O(n)$ space. This answers a longstanding open question from Dai, Martin, Robshaw, Wild [Cryptography and Coding III (1993)]. Our sequences are applied to find new longest-known orientable sequences for $n\leq 20$.

翻译：$n$ 阶可定向序列是一种循环二进制序列，其中每个长度为$n$的子串在\emph{任一方向上}至多出现一次。目前仅已知$n\leq 7$时最大长度的可定向序列，且其长度的平凡上界为$2^{n-1} - 2^{\lfloor(n-1)/2\rfloor}$。本文提出了首个高效算法，用于构造渐近最优长度的可定向序列；具体而言，我们的算法通过环接合与后继规则方法构造可定向序列，每个符号所需时间为$O(n)$，空间复杂度为$O(n)$。这解答了Dai、Martin、Robshaw、Wild在[Cryptography and Coding III (1993)]中提出的长期未决问题。应用我们的序列，对于$n\leq 20$的情形，发现了已知最长的可定向序列。