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 {Tiβ(H)}i=0nβ
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 Hβ be the parity-check matrix
of C which is formed by all the non-zero codewords of its dual code
Cβ₯. A parity-check matrix H is called BEC-optimal if
Tiβ(H)=Tiβ(Hβ),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.Comment: 33 pages, submitted to IEEE Trans. Inform. Theory, Feb. 201