Stopping Set Distributions of Some Reed–Muller Codes
Yong Jiang, Shu‐Tao Xia, Fang‐Wei Fu · IEEE Transactions on Information Theory · 2011
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). LetCbe a binary [n,k] linear code with parity-check matrixH, where the rows ofHmay be dependent. A stopping setSofCwith parity-check matrixHis a subset of column indices ofHsuch that the restriction ofHtoSdoes not contain a row of weight one. The stopping set distribution {Ti(H)}i=0nenumerates the number of stopping sets with sizeiofCwith parity-check matrixH. Note that stopping sets and stopping set distribution are related to the parity-check matrixHofC. LetH*be the parity-check matrix ofCwhich is formed by all the nonzero codewords of its dual codeC⊥. A parity-check matrixHis called BEC-optimal ifTi(H)=Ti(H*),i=0,1,...,nandHhas 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.