On balanced $(Z_{4u}\times Z_{8v},\{4,5\},1)$ difference packings

TL;DR

This paper determines the largest balanced difference packings in specific finite groups and derives optimal optical orthogonal signature pattern codes for those packings, advancing combinatorial design theory.

Contribution

It establishes the maximum size of balanced $(Z_{4u} imes Z_{8v},{4,5},1)$ difference packings for odd $uv$, and constructs corresponding optimal codes.

Findings

Largest possible balanced difference packings identified

Optimal optical orthogonal signature pattern codes constructed

Results applicable when $uv$ is odd

Abstract

Let $K$ be a set of positive integers and let $G$ be an additive group. A $(G, K, 1)$ difference packing is a set of subsets of $G$ with sizes from $K$ whose list of differences covers every element of $G$ at most once. It is balanced if the number of blocks of size $k\in K$ does not depend on $k$. In this paper, we determine a balanced $(Z_{4u}\times Z_{8v},{4,5},1)$ difference packing of the largest possible size whenever $uv$ is odd. The corresponding optimal balanced $(4u, 8v,\{4,5\},1)$ optical orthogonal signature pattern codes are also obtained.