Constructing Orientable Sequences

Abstract

This paper describes new, simple, recursive methods of construction for <i>orientable sequences</i>, i.e. periodic binary sequences in which any <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> consecutive encoded bits to determine its location and orientation on the surface. The only previously described method of construction (due to Dai <i>et al.</i>) is somewhat complex, whereas the new techniques are simple to both describe and implement. The methods of construction cover both the standard &#x2018;infinite periodic&#x2019; 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.