Stopping Set Distributions of Some Linear Codes

2006 IEEE Information Theory Workshop - ITW '06 Chengdu Pub Date : 2006-10-01 DOI:10.1109/ITW2.2006.323751

Yong Jiang, Shutao Xia, Fang-Wei Fu

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引用次数: 7

Abstract

In this paper, the stopping set distributions (SSD) of some well-known binary linear codes are determined by using finite geometry theory. Similar to the weight distribution of a binary linear code, the SSD {Ti(H)}n i=0 enumerates the number of stopping sets with size i of a linear code with parity-check matrix H. First, we deal with the simplex codes and Hamming codes. With parity-check matrix formed by all the weight 3 codewords of the Hamming code, the SSD of the simplex code is completely determined with explicit formula. With parity-check matrix formed by all the nonzero codewords of the simplex code, the SSD of the Hamming code is completely determined with two recursive equations. Then, the first order Reed-Muller codes and the extended Hamming codes are discussed. With parity-check matrix formed by all the weight 4 codewords of the extended Hamming code, the SSD of the first order Reed-Muller code is completely determined with explicit formula. With parity-check matrix formed by all the minimum codewords of the first order Reed-Muller code, the SSD of the extended Hamming code is completely determined with two recursive equations

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若干线性码的停止集分布

本文利用有限几何理论确定了一些著名二进制线性码的停止集分布。与二元线性码的权重分布类似，SSD {Ti(H)}n i=0枚举具有奇偶校验矩阵H的线性码的大小为i的停止集的个数。首先，我们处理单纯形码和汉明码。利用汉明码的所有权码字组成的奇偶校验矩阵，用显式公式完全确定了单纯形码的SSD。利用单纯形码的所有非零码字组成的奇偶校验矩阵，用两个递推方程完全确定了汉明码的SSD。然后，讨论了一阶Reed-Muller码和扩展汉明码。利用扩展汉明码的所有权码字组成的奇偶校验矩阵，用显式公式完全确定了一阶Reed-Muller码的SSD。利用由一阶Reed-Muller码的所有最小码字构成的奇偶校验矩阵，用两个递推方程完全确定了扩展汉明码的SSD

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2006 IEEE Information Theory Workshop - ITW '06 Chengdu

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