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 $n\leq 20$.

翻译：$n$阶可定向序列是一种循环二进制序列，其任意长度为$n$的子串在每个方向上至多出现一次。目前仅知$n\leq 7$时存在最大长度的可定向序列，其长度的一个平凡上界为$2^{n-1} - 2^{\lfloor(n-1)/2\rfloor}$。本文提出了首个高效构造渐近最优长度可定向序列的算法；具体而言，我们的算法通过环连接与后继规则方法构造可定向序列，每比特需要$O(n)$时间与$O(n)$空间。这解答了Dai、Martin、Robshaw、Wild在《Cryptography and Coding III (1993)》中提出的长期开放问题。应用最新的级联树框架，相同序列可在每比特$O(1)$摊销时间内生成，使用$O(n^2)$空间。我们将所得序列应用于寻找$n\leq 20$时新的最长已知（非周期）可定向序列。