Abstract
A \((3,4)\)-packing 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 \((3,4)\)-packing, 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 \(J(n,4,4)\,(=\lfloor \frac{n}{4}\lfloor \frac{n-1}{3}\lfloor \frac{n-2}{2}\rfloor \rfloor \rfloor )\).
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The authors would like to thank the referee for many helpful comments. Research supported by NSFC Grant 11222113.
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Communicated by T. Etzion.
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Bao, J., Ji, L. The completion determination of optimal \((3,4)\)-packings. Des. Codes Cryptogr. 77, 217–229 (2015). https://doi.org/10.1007/s10623-014-0001-2
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DOI: https://doi.org/10.1007/s10623-014-0001-2