A remark on Plotkin's bound

Let A(n, d) denote the greatest number of codewords possible in a binary bl ock code of length n and distance d. Plotkin gave a simple counting argumen t which leads to an upper bound B(n, d) for A(n, d) when d greater than or equal to n/2. Levenshtein proved that if Hadamard's conjecture is true then Plotkin's bound is sharp. Though Hadamard's conjecture is probably true, i ts resolution remains a difficult open question. So it is natural to ask wh at one can prove about the ratio R(n, d) = A(n, d)/B(n, d). This note prese nts an efficient heuristic for constructing, for any d greater than or equa l to n/2, a binary code which has at least 0.495B(n, d) codewords. A comput er calculation confirms that R(n, d) > 0.495 for d up to one trillion.