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A remark on Plotkin's bound

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2 Author(s)
W. de Launey ; IDA Center for Commun. Res., San Diego, CA, USA ; D. M. Gordon

Let A(n,d) denote the greatest number of codewords possible in a binary block code of length n and distance d. Plotkin gave a simple counting argument which leads to an upper bound B(n,d) for A(n,d) when d>n/2. Levenshtein (1964) proved that if Hadamard's conjecture is true then Plotkin's bound is sharp. Though Hadamard's conjecture is probably true, its resolution remains a difficult open question. So it is natural to ask what one can prove about the ratio R(n,d)=A(n,d)/B(n,d). This note presents an efficient heuristic for constructing, for any d⩾n/2, a binary code which has at least 0.495B(n,d) codewords. A computer calculation confirms that R(n,d)>0.495 for d up to one trillion

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IEEE Transactions on Information Theory  (Volume:47 ,  Issue: 1 )