Some bounds for error-correcting codes

Abstract

This paper concerns itself with binary coding and more specifically with an upper bound for the number of binary code works of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> such that every two differ in at least <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> positions from each other. It is shown that there is a relationship between this problem and a well-known extremum problem. The techniques developed for the latter problem are used to derive upper bounds for the former, one of which has been obtained in an entirely different way by Plotkin. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B(n, d)</tex> denotes the maximum number of binary code words of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> and distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> , it is shown that \begin{equation} B(n,d) \leq \frac{2d}{2d-n} for d > n/2. \end{equation} The second bound due to Rankin is \begin{equation} B(n, d) \leq \frac{8d(n - d)}{n - (n- 2d)^2} for n - \sqrt{n} < 2d \leq n \end{equation} provided the binary code words consist of pairs, each pair differing in all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> positions. Although Rankin's bound contains the restriction that the binary code words consist of pairs, each pair differing in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> positions, it is shown that the restriction may not be severe in the sense that the bound can often be attained.