Negative Avoiding Sequences

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 $k\ge 3$ and every $n\ge 2$.