New upper bounds for the period of a negative orientable sequence
Chris J Mitchell, Peter R Wild
TL;DR
This paper studies negative orientable sequences NOS_k(n) and derives sharper upper bounds on their period by translating combinatorial constraints into a de Bruijn graph framework. The authors define the nega-sequence-subgraph in the de Bruijn graph and exploit in-degree, out-degree, and parity constraints of vertices (linked to left-sns, right-sns, and negasymmetric tuples) to bound the total number of admissible edges, hence bounding the period via an Eulerian-circuit argument. A key contribution is the explicit, parity-sensitive bound on the NOS period for all $n$ and $k$, including detailed formulas for the challenging cases $n=2,3,4$ and the general $n>4$ regime. While these results sharpen previous bounds significantly for $n>2$, the paper notes a remaining gap between known largest periods and the theoretical bounds, indicating ongoing open problems and opportunities for computation-assisted advances.
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 are significantly sharper then the previous known bound for n>2. 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.