Special orientable sequences

TL;DR

This paper introduces methods to construct special orientable sequences over arbitrary alphabets, extending previous work and enabling applications in position-location with sequences that have near-optimal periods.

Contribution

The paper develops new construction methods for special orientable sequences applicable to any alphabet size and window length, broadening their practical utility.

Findings

Sequences constructed have asymptotically optimal periods as alphabet size increases.

Methods work for all alphabet sizes larger than a small lower bound.

Sequences are suitable for automatic position-location applications.

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., recursive methods of construction were described for orientable sequences over arbitrary finite alphabets, requiring 'starter sequences' with special properties. Some of these methods required as input special orientable sequences, i.e. orientable sequences which were simultaneously negative orientable. We exhibit methods for constructing special orientable sequences with properties appropriate for use in two of the recursive methods of Alhakim et al. As a result we are able to show how to construct special orientable sequences for arbitrary sizes of alphabet (larger than a small lower bound) and for all window sizes. These sequences have periods asymptotic to the optimal as the alphabet size increases.