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

TL;DR

The paper tackles the problem of constructing orientable sequences $\,\mathcal{OS}(n)\,$ of asymptotically optimal length $L_n$, addressing a long-standing open question. It introduces an efficient cycle-joining construction based on asymmetric bracelets $\mathbf{A}(n)$ and a carefully designed successor rule that outputs each bit in $O(n)$ time using $O(n)$ space, producing an OS of length $L_n$. By applying concatenation-tree theory, the same sequences can be generated in $O(1)$-amortized time per bit with $O(n^2)$ space, enabling practical generation for larger $n$ (up to 20 shown). The work also extends to longer and acyclic orientable sequences via extensions and recursive lifts, yielding the longest-known $AOS(n)$ for $n\le 20$ and providing implementations. These results advance both the theoretical bounds and practical construction of orientable sequences, with potential applications in robotic sensing and sequence design.

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 $n\leq 20$.

Figures (3)

• Figure 1: Cycle-joining trees for $\mathbf{B}(6)$ from simple parent rules.
• Figure 2: The cycle-joining tree $\mathbb{T}_9$. The black edges indicate that $\mathop{\mathrm{par}}\nolimits(\alpha) = \mathop{\mathrm{first1}}\nolimits(\alpha)$; the blue edges indicate that $\mathop{\mathrm{par}}\nolimits(\alpha) = \mathop{\mathrm{last1}}\nolimits(\alpha)$; the red edges indicate that $\mathop{\mathrm{par}}\nolimits(\alpha) = \mathop{\mathrm{last0}}\nolimits(\alpha)$.
• Figure 3: The concatenation tree $\mathcal{T}_9$ derived from the cycle-joining tree $\mathbb{T}_9$ shown in Figure \ref{['fig:cycle9']}. The small grey box on the top edge of each node indicates the change index; the left-children descend from blue dots $\bullet$ and the right-children descend from red dots $\bullet$. The small numbers above each node indicate the order the nodes are visited in an RCL traversal.

Theorems & Definitions (14)

• Theorem 1
• Lemma 2
• Corollary 3
• Remark 4
• Proposition 5
• Theorem 6
• Lemma 7
• Theorem 8
• Lemma 9
• Lemma 10