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：设$n$阶可定向序列是一个循环二进制序列，其中每个长度为$n$的子串在任意方向上至多出现一次。已知最大长度的可定向序列仅适用于$n\leq 7$，且其长度的平凡上界为$2^{n-1} - 2^{\lfloor(n-1)/2\rfloor}$。本文提出首个高效算法，可构造长度渐近最优的可定向序列；具体而言，我们的算法通过循环连接和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$O(n)$。这回答了Dai、Martin、Robshaw、Wild [Cryptography and Coding III (1993)]中一个长期悬而未决的开放问题。我们将所构造的序列应用于寻找$n\leq 20$时新的最长已知可定向序列。