New orientable sequences

TL;DR

This paper improves bounds on the period of orientable sequences, introduces a new construction method using de Bruijn graph subgraphs, and produces sequences with larger periods than previously known for certain parameters.

Contribution

It provides improved upper bounds and a novel construction approach for orientable sequences, achieving larger periods for specific sequence orders and alphabet sizes.

Findings

Sequences with larger periods for n=2,3,4, and odd k

New construction method using de Bruijn graph subgraphs

Sequences surpassing previous period records for 4≤n≤8

Abstract

Orientable sequences of order n are infinite periodic sequences with symbols drawn from a finite alphabet of size k with the property that any particular subsequence of length n occurs at most once in a period in either direction. They were introduced in the early 1990s in the context of possible applications in position sensing. Bounds on the period of such sequences and a range of methods of construction have been devised, although apart from very small cases a significant gap remains between the largest known period for such a sequence and the best known upper bound. In this paper we first give improved upper bounds on the period of such sequences. We then give a new general method of construction for orientable sequences involving subgraphs of the de Bruijn graph with special properties, and describe two different approaches for generating such subgraphs. This enables us to construct orientable sequences with periods meeting the improved upper bounds when n is 2 or 3, as well as n=4 and k odd. For 4\leq n\leq 8, in some cases the sequences produced by the methods described have periods larger than for any previously known sequences.