Determination of sizes of optimal three-dimensional optical orthogonal codes of weight three with the AM-OPP restriction

TL;DR

This paper determines the exact sizes of optimal three-dimensional optical orthogonal codes with weight three under the AM-OPP restriction, using new combinatorial design constructions for all positive parameters.

Contribution

It introduces new combinatorial designs and provides a complete determination of the code sizes for all positive parameters in the specified class.

Findings

Exact number of codewords for optimal codes is determined.

New auxiliary designs facilitate the constructions.

Results apply to all positive integers u, v, w with u ≥ 3.

Abstract

In this paper, we further investigate the constructions on three-dimensional $(u\times v\times w,k,1)$ optical orthogonal codes with the at most one optical pulse per wavelength/time plane restriction (briefly AM-OPP $3$-D $(u\times v\times w,k,1)$-OOCs) by way of the corresponding designs. Several new auxiliary designs such as incomplete holey group divisible designs and incomplete group divisible packings are introduced and therefore new constructions are presented. As a consequence, the exact number of codewords of an optimal AM-OPP $3$-D $(u\times v\times w,3,1)$-OOC is finally determined for any positive integers $v,w$ and $u\geq3$.