Orientable and negative orientable sequences

TL;DR

This paper introduces three new methods for generating negative orientable sequences over finite alphabets, providing bounds on their periods and demonstrating their use in constructing nearly maximal orientable sequences for various alphabet sizes.

Contribution

It presents novel techniques for creating negative orientable sequences and applies them to generate nearly optimal orientable sequences across different alphabet sizes.

Findings

Three techniques for generating negative orientable sequences.

Upper bounds established for the periods of these sequences.

Construction of orientable sequences close to maximum period for all non-binary alphabets.

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.