Maximum $w$-cyclic holely group divisible packings with block size three and applications to optical orthogonal codes

TL;DR

This paper develops combinatorial constructions for maximum $w$-cyclic holely group divisible packings with block size three, and applies these results to determine the size of optimal optical orthogonal codes in three dimensions.

Contribution

It provides exact counts for maximum $w$-cyclic 3-HGDPs of certain types and applies these to optimize three-dimensional optical orthogonal codes.

Findings

Exact number of base blocks for maximum $w$-cyclic 3-HGDPs of type $(u,w^v)$.

Determination of the maximum codewords in 3D optical orthogonal codes.

Applications to optical code design with specific constraints.

Abstract

In this paper we investigate combinatorial constructions for $w$-cyclic holely group divisible packings with block size three (briefly by $3$-HGDPs). For any positive integers $u,v,w$ with $u\equiv0,1~(\bmod~3)$, the exact number of base blocks of a maximum $w$-cyclic $3$-HGDP of type $(u,w^v)$ is determined. This result is used to determine the exact number of codewords in a maximum three-dimensional $(u\times v\times w,3,1)$ optical orthogonal code with at most one optical pulse per spatial plane and per wavelength plane.