A remark on Plotkin's bound
نویسندگان
چکیده
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 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. Keywords— Plotkin bound, Hadamard matrix, Paley matrix, Goldbach conjecture, high distance binary block codes I. Preliminaries and Overview For n > d > 0, let A(n, d) denote the maximum number of codewords possible in a binary block code of length n and minimum (Hamming) distance d. Notice that, if d is odd, then C is an (n,M, d) code if and only if the code C obtained by adding a parity check bit to each codeword in C is an (n + 1,M, d + 1) code. Therefore, if d is even, then A(n, d) = A(n− 1, d− 1). So in order to understand the behaviour of A(n, d) it is sufficient to understand its behaviour for d even. An elementary counting argument gives Plotkin’s bound. This states that for d even, A(n, d) ≤ B(n, d) = {
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عنوان ژورنال:
- IEEE Trans. Information Theory
دوره 47 شماره
صفحات -
تاریخ انتشار 2001