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Constructions of two-dimensional optical orthogonal codes of weight three

Xiuling Shan, Lidong Wang, Yanxun Chang, Xiaomiao Wang

TL;DR

The work advances two-dimensional optical orthogonal codes by focusing on weight-3, λ=1 codes and delivering a unified suite of set-theoretic descriptions, improved bounds, and recursive constructions. It leverages holey designs (HGDDs, IHGDDs, SCHGDDs), n-cyclic GDPs, and Skolem/Langford-type sequences to build large, provably optimal 2-D OOCs and to determine Φ$(m\times n,3,1)$ across broad parameter regimes. The main theorem provides explicit deficits μ under several modular conditions, while auxiliary designs enable scalable inflation from small base blocks to large grids. The practical impact lies in enabling efficiently constructible, high-size 2-D OOCs suitable for OCDMA systems, with concrete constructions for many parameter sets.

Abstract

The study of optical orthogonal codes has been motivated by an application in an optical code-division multiple access system. This paper focuses on optimal two-dimensional optical orthogonal codes with autocorrelation and cross-correlation both equal to $1$. By examining the structures of $n$-cyclic group divisible packings and semi-cyclic incomplete holey group divisible designs, we present new combinatorial constructions for two-dimensional $(m\times n,k,1)$-optical orthogonal codes. As a consequence, the exact number of codewords of an optimal two-dimensional $(m\times n,3,1)$-optical orthogonal code is determined for any positive integers $m$ and $n$.

Constructions of two-dimensional optical orthogonal codes of weight three

TL;DR

The work advances two-dimensional optical orthogonal codes by focusing on weight-3, λ=1 codes and delivering a unified suite of set-theoretic descriptions, improved bounds, and recursive constructions. It leverages holey designs (HGDDs, IHGDDs, SCHGDDs), n-cyclic GDPs, and Skolem/Langford-type sequences to build large, provably optimal 2-D OOCs and to determine Φ across broad parameter regimes. The main theorem provides explicit deficits μ under several modular conditions, while auxiliary designs enable scalable inflation from small base blocks to large grids. The practical impact lies in enabling efficiently constructible, high-size 2-D OOCs suitable for OCDMA systems, with concrete constructions for many parameter sets.

Abstract

The study of optical orthogonal codes has been motivated by an application in an optical code-division multiple access system. This paper focuses on optimal two-dimensional optical orthogonal codes with autocorrelation and cross-correlation both equal to . By examining the structures of -cyclic group divisible packings and semi-cyclic incomplete holey group divisible designs, we present new combinatorial constructions for two-dimensional -optical orthogonal codes. As a consequence, the exact number of codewords of an optimal two-dimensional -optical orthogonal code is determined for any positive integers and .
Paper Structure (27 sections, 57 theorems, 25 equations)

This paper contains 27 sections, 57 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Set-theoretic description
  3. Improved upper bounds for weight three
  4. $2$-D $(m\times n,k,1)$-OOCs with "holes"
  5. $2$-D $([m:r]\times n,k,1)$-OOCs
  6. $g$-regular $2$-D $([m:r]\times n,k,1)$-OOCs
  7. More examples of $g$-regular $2$-D $([m:r]\times n,k,1)$-OOCs
  8. Auxiliary designs
  9. $n$-cyclic group divisible packings
  10. $wt$-cyclic holey group divisible designs
  11. Semi-cyclic incomplete holey group divisible designs
  12. Constructions for $g$-regular $2$-D $([m:r]\times n,k,1)$-OOCs
  13. Constructions for $2$-D $([m:r]\times n,k,1)$-OOCs
  14. Skolem-type sequences
  15. Sporadic recursive constructions
  16. ...and 12 more sections

Key Result

Lemma 1.2

yk$\Phi(m\times n,k,\lambda)\leqslant J(m\times n,k,\lambda)$, where

Theorems & Definitions (101)

  • Definition 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 91 more