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Negative Avoiding Sequences

Chris J Mitchell, Peter R Wild

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 $n\geq2$.

Negative Avoiding Sequences

Abstract

Negative avoiding sequences of span are periodic sequences of elements from for some with the property that no -tuple occurs more than once in a period and if an -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 and every .

Paper Structure

This paper contains 17 sections, 18 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Motivation and related work
  3. The de Bruijn graph and the Lempel homomorphism
  4. The de Bruijn graph
  5. Antinegative subgraphs
  6. Implications
  7. The Lempel Homomorphism
  8. A simple bound and some examples
  9. Maximal sequences for odd n
  10. Preliminaries
  11. The construction
  12. The n even case is not so simple
  13. Maximal sequences for n=2
  14. Maximal sequences for n=2 and k odd
  15. Maximal sequences for n=2 and k even
  16. ...and 2 more sections

Key Result

Lemma 3.1

Suppose $S$ is a $k$-ary periodic $n$-window sequence. Then $S$ is an $\mathcal{NAS}_k(n)$ if and only if $E_n(S)$ is antinegative in $B_{k}(n-1)$.

Theorems & Definitions (52)

  • Definition 1.1: Alhakim24a
  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4: Definition 3.2 of Mitchell26
  • Lemma 3.1
  • proof
  • ...and 42 more