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首页> 外文期刊>Discrete Applied Mathematics >On the combinatorial structure of a class of [(m 2), (m-1 2), 3] shortened Hamming codes and their dual-codes
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On the combinatorial structure of a class of [(m 2), (m-1 2), 3] shortened Hamming codes and their dual-codes

机译：关于[[m 2），（m-1 2），3]短汉明码及其对偶码的组合结构

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Let H-m be the binary linear block code with parity-check matrix H-m whose columns are all distinct binary strings of length m and Hamming weight 2. It is shown that H-m is an [n, k, d] = [m(m-1)/2, (m-1)(m-2)/2, 3] code while the dual-code H-m(perpendicular to) has dimension k(perpendicular to) and minimum distance d(perpendicular to) satisfying k(perpendicular to) = d(perpendicular to) = m - 1. It is in general very difficult to find or even estimate the covering radius of a given code. It is shown here that the covering radius of H-m, denoted Cr(H-m), is left perpendicularm/2right perpendicular. We also show that Cr(H-m(perpendicular to)) = m(m-2)/4 if m is even and Cr(H-m(perpendicular to)) = (m-1)(2)/4 if m is odd. Thus Cr(H-m(perpendicular to)) similar or equal to Cr(H-m)(2). The weight distribution of H-m(perpendicular to) is given. This together with the MacWilliams identities results in an expression for the weight distribution of H-m. It turns out that the covering radius of H-m is equal to its external distance. From the Tanner graph perspective, the Tanner graphs of H-m and H-m(perpendicular to) have girth 6. It is shown that the Tanner graphs of H-m+1(perpendicular to) and H-m are essentially identical and are structurally representable by the complete graph K-m on m vertices. (C) 2008 Elsevier B.V. All rights reserved.
机译：设Hm为带有奇偶校验矩阵Hm的二进制线性块代码，其列均为长度为m且汉明权重为2的不同二进制字符串。表明Hm为[n，k，d] = [m（m-1） ）/ 2，（m-1）（m-2）/ 2，3]码，而双重码Hm（垂直于）具有尺寸k（垂直于）和最小距离d（垂直于）满足k（垂直于）= d（垂直于）= m-1。通常很难找到甚至估计给定代码的覆盖半径。在此示出，H-m的覆盖半径表示为Cr（H-m），为左垂直m / 2右垂直。我们还表明，如果m为偶数，则Cr（H-m（垂直于））= m（m-2）/ 4;如果m为奇数，则Cr（H-m（垂直于））=（m-1）（2）/ 4。因此，Cr（H-m（垂直于））类似于或等于Cr（H-m）（2）。给出了H-m（垂直于）的重量分布。这与MacWilliams身份一起可表示H-m的重量分布。事实证明，H-m的覆盖半径等于其外部距离。从Tanner图的角度来看，Hm和Hm（垂直于）的Tanner图的围长为6。表明，H-m + 1（垂直于）和Hm的Tanner图基本相同，并且在结构上可以用整数表示。在m个顶点上绘制图Km。 （C）2008 Elsevier B.V.保留所有权利。

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