On the Packing of Pairs by Quintuples With Index $2$

Abstract

A packing design (briefly packing) of order $v$, block size $k$, and index $\lambda$ is a pair $(X,\mathcal{D})$ where $X$ is a $v$-set (of points) and $\mathcal{D}$ is a collection of $k$-subsets of $X$ (called blocks) with cardinality $b$ such that every $2$-subset of $X$ is contained in at most $\lambda$ blocks of $\mathcal{D}$. We denote it by $\mathrm{S}\mathrm{D}(k,\lambda ;v,b)$. If no other such packing has more blocks, the packing is said to be maximum, and the number of blocks in $\mathcal{D}$ is the packing number $\mathrm{D}(k,\lambda ;v)$. For fixed $k$,$\lambda$ and $v$, the packing problem is to determine the packing number. In this paper, the values of $\mathrm{D}(5,2;v)$ are determined for all $v\ge 5$ except $48$ values of $v$.