Constructing orientable sequences

TL;DR

This paper introduces simple, recursive methods for constructing orientable binary sequences with unique n-tuple occurrences, enhancing applications in position-location systems with easier implementation.

Contribution

The paper presents novel, straightforward recursive techniques for creating orientable sequences, extending previous complex methods and covering both infinite and finite sequence cases.

Findings

Methods are simple to describe and implement.

Techniques build on Lempel homomorphism for recursive generation.

Applicable to both infinite and finite sequences.

Abstract

This paper describes new, simple, recursive methods of construction for orientable sequences, i.e. periodic binary sequences in which any n-tuple occurs at most once in a period in either direction. As has been previously described, such sequences have potential applications in automatic position-location systems, where the sequence is encoded onto a surface and a reader needs only examine n consecutive encoded bits to determine its location and orientation on the surface. The only previously described method of construction (due to Dai et al.) is somewhat complex, whereas the new techniques are simple to both describe and implement. The methods of construction cover both the standard `infinite periodic' case, and also the aperiodic, finite sequence, case. Both the new methods build on the Lempel homomorphism, first introduced as a means of recursively generating de Bruijn sequences.