Stopping Set Distributions of Some Reed–Muller Codes

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=0n 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.