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

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.