Stopping Set Distributions of Some Linear Codes

Stopping sets and stopping set distribution of an low-density parity-check 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 \{T_i(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 non-zero codewords of its dual code C^{\perp}. A parity-check matrix H is called BEC-optimal if T_i(H)=T_i(H^*), i=0,1,..., n and H has the smallest number of rows. On the BEC, iterative decoder of C with BEC-optimal parity-check matrix is an optimal decoder with much lower decoding complexity than the exhaustive decoder. 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.