Negative Avoiding Sequences

TL;DR

This paper introduces negative avoiding sequences, a special class of periodic sequences with unique properties, establishes an upper bound on their period, and proves their existence for all relevant parameters.

Contribution

It defines negative avoiding sequences, derives an upper bound on their period, and proves their existence for all integers k ≥ 3 and n ≥ 2.

Findings

Established a simple upper bound on the period of negative avoiding sequences.

Proved the existence of maximal negative avoiding sequences for all k ≥ 3 and n ≥ 2.

Abstract

Negative avoiding sequences of span $n$ are periodic sequences of elements from $\mathbb{Z}_k$ for some $k$ with the property that no $n$-tuple occurs more than once in a period and if an $n$-tuple does occur then its negative does not. They are a special type of cut-down de Bruijn sequence with potential position-location applications. We establish a simple upper bound on the period of such a sequence, and refer to sequences meeting this bound as maximal negative avoiding sequences. We then go on to demonstrate the existence of maximal negative avoiding sequences for every $k\geq3$ and every $n\geq2$.