New upper bounds for the period of a negative orientable sequence

TL;DR

This paper establishes significantly improved upper bounds on the period length of negative orientable sequences, which are special periodic sequences with unique n-tuple properties, using graph-theoretic analysis.

Contribution

It introduces sharper upper bounds for the period of negative orientable sequences for n>2, advancing understanding of their structural limitations.

Findings

New upper bounds are significantly sharper than previous bounds.

The bounds are derived through analysis of the de Bruijn graph substructure.

A gap remains between known sequences and theoretical bounds for n>2.

Abstract

Negative orientable sequences, i.e. periodic sequences with elements from a finite alphabet of size at least three in which an n-tuple or the negative of its reverse appears at most once in a period of the sequence, were introduced by Alhakim et al. in 2024. The main goal in defining them was as a means of generating orientable sequences, which have automatic position location applications, although they are potentially of interest in their own right. In this paper we develop new upper bounds on the period of negative orientable sequences which, for n>2, are significantly sharper than the previous known bound. The approach used to develop the new bounds involves examining the nodes in the subgraph of the de Bruijn graph corresponding to a negative orientable sequence, and to consider the implications of the fact that the in-degree of every vertex in this subgraph must equal the out-degree. However, despite improving the bounds, a gap remains between the largest known period for a negative orientable sequence and the corresponding bounds for every n>2.