The completion of optimal (3,4)-packings

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A 3-(n,4,1) packing design consists of an n-element set X and a collection of 4-element subsets of X, called 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) (=/4-1/3-2/2) where n=6k+5, k∈ {m:m is odd, 3≤ m≤ 35,m≠ 17,21}∪ {45,47,75,77,79,159}.