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Constructing orientable and negative orientable sequences with asymptotically optimal period

Chris J Mitchell, Peter R Wild

Abstract

Orientable sequences, periodic sequences in which any $n$-tuple appears at most once in either direction, were introduced in the early 1990s for use in certain position location applications; constructions and upper bounds on the period for the binary case were published by Dai et al. More recent work has focussed on $k$-ary sequences for arbitrary $k>2$; one method of construction involves negative orientable sequences, in which an $n$-tuple appears at most once in either the sequence or the negative of its reverse. In this paper we show how additional $n$-tuples can be added to one previously described approach for generating negative orientable sequences, resulting in new sequences with asymptotically optimal period. These sequences can in turn be used to generate orientable sequences, again with asymptotically optimal period.

Constructing orientable and negative orientable sequences with asymptotically optimal period

Abstract

Orientable sequences, periodic sequences in which any -tuple appears at most once in either direction, were introduced in the early 1990s for use in certain position location applications; constructions and upper bounds on the period for the binary case were published by Dai et al. More recent work has focussed on -ary sequences for arbitrary ; one method of construction involves negative orientable sequences, in which an -tuple appears at most once in either the sequence or the negative of its reverse. In this paper we show how additional -tuples can be added to one previously described approach for generating negative orientable sequences, resulting in new sequences with asymptotically optimal period. These sequences can in turn be used to generate orientable sequences, again with asymptotically optimal period.
Paper Structure (18 sections, 29 theorems, 76 equations, 4 tables)

This paper contains 18 sections, 29 theorems, 76 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Motivation and structure
  4. Notation and basic results
  5. Partitioning $H_k(n-1)$ into circuits
  6. Defining the partition
  7. Examples
  8. Properties of circuits in $\mathcal{C}_k(n-1)$
  9. Adding circuits from $\mathcal{C}_k(n-1)$ to $E_k(n-1)$
  10. Enumerating non-negasymmetric circuits
  11. Preliminaries
  12. The $n$ odd case
  13. The $n$ even case
  14. The main result
  15. Constructing negative orientable and orientable sequences
  16. ...and 3 more sections

Key Result

Lemma 1.1

Suppose $k\geq 3$ and $n\geq3$. The number of negasymmetric $n$-tuples is

Theorems & Definitions (76)

  • Definition 1.1: Alhakim24a
  • Definition 1.2: Alhakim24a
  • Definition 1.3: Alhakim24a
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 66 more