Stopping Set Distributions of Some Reed&amp;#x2013;Muller Codes

Yong Jiang,Fang-Wei Fu,Shu-Tao Xia

doi:10.1109/tit.2011.2162181

Abstract

Stopping sets and stopping set distribution of a linear code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let C be a binary [ n , k ] linear code with parity-check matrix H , where the rows of H may be dependent. A stopping set S of C with parity-check matrix H is a subset of column indices of H such that the restriction of H to S does not contain a row of weight one. The stopping set distribution { Ti ( H )} i =0 n enumerates the number of stopping sets with size i of C with parity-check matrix H . Note that stopping sets and stopping set distribution are related to the parity-check matrix H of C . Let H * be the parity-check matrix of C which is formed by all the nonzero codewords of its dual code C ⊥ . A parity-check matrix H is called BEC-optimal if Ti ( H )= Ti ( H * ), i =0,1,..., n and H has the smallest number of rows. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes, and the extended Hamming codes, which are some Reed-Muller codes or their shortening or puncturing versions.

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