The completion of optimal $(3,4)$-packings

Abstract

A 3-$(n,4,1)$ packing design consists of an $n$-element set $X$ and a collection of $4$-element subsets of $X$, called {\it blocks}, such that every $3$-element subset of $X$ is contained in at most one block. The packing number of quadruples $d(3,4,n)$ denotes the number of blocks in a maximum $3$-$(n,4,1)$ packing design, which is also the maximum number $A(n,4,4)$ of codewords in a code of length $n$, constant weight $4$, and minimum Hamming distance 4. In this paper the undecided 21 packing numbers $A(n,4,4)$ are shown to be equal to Johnson bound $J(n,4,4)$ $(=\lfloor \frac{n}{4}\lfloor \frac{n-1}{3}\lfloor \frac{n-2}{2}\rfloor \rfloor \rfloor )$ where $n=6k+5$, $k\in \{m:m$ is odd, $3\le m\le 35,m\ne 17,21\}\cup \{45,47,75,77,79,159\}$.