Orientable and negative orientable sequences
Chris J Mitchell, Peter R Wild
TL;DR
This work advances the construction of orientable sequences over arbitrary alphabets by leveraging negative orientable sequences as building blocks. It presents three distinct NOS constructions, derives sharp period bounds, and demonstrates how to lift NOS to OS via the Lempel–Homomorphism on de Bruijn graphs, achieving near-optimal periods for various $n$ and $q$. A second recursive framework further yields long orientable sequences with systematic period growth. Together, these methods extend the nonbinary theory of orientable sequences and enable practical near-maximum-length constructions for position-resolution applications, while highlighting open questions on space efficiency and further generalizations.
Abstract
Analogously to de Bruijn sequences, orientable sequences have application in automatic position-location applications and, until recently, studies of these sequences focused on the binary case. In recent work by Alhakim et al., a range of methods of construction were described for orientable sequences over arbitrary finite alphabets; some of these methods involve using negative orientable sequences as a building block. In this paper we describe three techniques for generating such negative orientable sequences, as well as upper bounds on their period. We then go on to show how these negative orientable sequences can be used to generate orientable sequences with period close to the maximum possible for every non-binary alphabet size and for every tuple length. In doing so we use two closely related approaches described by Alhakim et al.