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An orientable sequence of order $n$ is a cyclic binary sequence such thateach length-$n$ substring appears at most once \emph{in either direction}.Maximal length orientable sequences are known only for $n\leq 7$, and a trivialupper bound on their length is $2^{n-1} - 2^{\lfloor(n-1)/2\rfloor}$. Thispaper presents the first efficient algorithm to construct orientable sequenceswith asymptotically optimal length; more specifically, our algorithm constructsorientable sequences via cycle-joining and a successor-rule approach requiring$O(n)$ time per bit and $O(n)$ space. This answers a longstanding open questionfrom Dai, Martin, Robshaw, Wild [Cryptography and Coding III (1993)]. Applyinga 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 appliedto find new longest-known (aperiodic) orientable sequences for $n\leq 20$.

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