Constructions of two-dimensional optical orthogonal codes of weight three

TL;DR

This paper develops new combinatorial methods to construct optimal two-dimensional optical orthogonal codes with specific autocorrelation and cross-correlation properties, crucial for optical communication systems.

Contribution

It introduces novel combinatorial constructions based on group divisible packings and designs, determining the exact number of codewords for optimal codes.

Findings

Exact number of codewords for optimal codes determined

New combinatorial constructions for 2D optical orthogonal codes

Codes have autocorrelation and cross-correlation both equal to 1

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 $n$.