A remark on Plotkin´s bound

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