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Optical orthogonal codes from a combinatorial perspective

Sophie Huczynska, Siaw-Lynn Ng

TL;DR

Optical orthogonal codes (OOCs) are sets of binary sequences with bounded auto- and cross-correlation; this paper decouples the two correlations by treating $\lambda_a$ and $\lambda_c$ separately and translating OOCs to subsets of $\mathbb{Z}_v$ to study internal/external differences. It surveys definitions, properties, and constructions for cases with $\lambda_a \neq \lambda_c$, establishes bounds on each parameter, and links OOCs to difference families, packings, and related combinatorial objects. The work also highlights new connections and open questions, including generalizations to arbitrary groups and equivalence vs isomorphism issues. These results broaden the design space for optical CDMA and provide structural insights into how decoupled correlation constraints shape OOC constructions.

Abstract

Optical orthogonal codes (OOCs) are sets of $(0,1)$-sequences with good auto- and cross-correlation properties. They were originally introduced for use in multi-access communication, particularly in the setting of optical CDMA communications systems. They can also be formulated in terms of families of subsets of $\mathbb{Z}_v$, where the correlation properties can be expressed in terms of conditions on the internal and external differences within and between the subsets. With this link there have been many studies on their combinatorial properties. However, in most of these studies it is assumed that the auto- and cross-correlation values are equal; in particular, many constructions focus on the case where both correlation values are $1$. This is not a requirement of the original communications application. In this paper, we "decouple" the two correlation values and consider the situation with correlation values greater than $1$. We consider the bounds on each of the correlation values, and the structural implications of meeting these separately, as well as associated links with other combinatorial objects. We survey definitions, properties and constructions, establish some new connections and concepts, and discuss open questions.

Optical orthogonal codes from a combinatorial perspective

TL;DR

Optical orthogonal codes (OOCs) are sets of binary sequences with bounded auto- and cross-correlation; this paper decouples the two correlations by treating and separately and translating OOCs to subsets of to study internal/external differences. It surveys definitions, properties, and constructions for cases with , establishes bounds on each parameter, and links OOCs to difference families, packings, and related combinatorial objects. The work also highlights new connections and open questions, including generalizations to arbitrary groups and equivalence vs isomorphism issues. These results broaden the design space for optical CDMA and provide structural insights into how decoupled correlation constraints shape OOC constructions.

Abstract

Optical orthogonal codes (OOCs) are sets of -sequences with good auto- and cross-correlation properties. They were originally introduced for use in multi-access communication, particularly in the setting of optical CDMA communications systems. They can also be formulated in terms of families of subsets of , where the correlation properties can be expressed in terms of conditions on the internal and external differences within and between the subsets. With this link there have been many studies on their combinatorial properties. However, in most of these studies it is assumed that the auto- and cross-correlation values are equal; in particular, many constructions focus on the case where both correlation values are . This is not a requirement of the original communications application. In this paper, we "decouple" the two correlation values and consider the situation with correlation values greater than . We consider the bounds on each of the correlation values, and the structural implications of meeting these separately, as well as associated links with other combinatorial objects. We survey definitions, properties and constructions, establish some new connections and concepts, and discuss open questions.

Paper Structure

This paper contains 23 sections, 27 theorems, 17 equations, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and fundamental properties
  3. OOCs as sequences
  4. OOCs as subsets
  5. Equivalence of OOCs
  6. Survey of bounds and constructions
  7. Bounds on the size of $(v,w, \lambda_a, \lambda_c)$-OOCs
  8. Some construction techniques with $\lambda_a \neq \lambda_c$
  9. Constructions using finite geometry
  10. Constructions using number theory
  11. Constructions using binary sequences
  12. Computer searches
  13. Recursive constructions
  14. Some constructions with $\lambda_a=\lambda_c >1$
  15. Relationship with other combinatorial objects
  16. ...and 8 more sections

Key Result

Lemma 2.3

A $(v,w,\lambda_a, \lambda_c)$-OOC has $\lambda_a=0$ if and only if $w=1$, that is, all the sequences have exactly one 1 and $v-1$ 0s.

Theorems & Definitions (66)

  • Definition 2.1: OOCs as sequences
  • Remark 2.2: The case $N=1$
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5: Non-trivial OOCs
  • Remark 2.6: On correlation
  • Definition 2.7: Proper OOCs
  • Definition 2.8: Full cyclic order
  • Definition 2.9: Non-degenerate OOCs
  • Theorem 2.10
  • ...and 56 more