ISSN 0032-9460, Problems of Information Transmission, 2006, Vol. 42, No. 2, pp. 106–113.
Pleiades Publishing, Inc., 2006.
Original Russian Text
D.K. Zigangirov, K.Sh. Zigangirov, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 2, pp. 44–52.
Decoding of Low-Density Codes with Parity-Check
Matrices Composed of Permutation Matrices
in an Erasure Channel
D. K. Zigangirov and K. Sh. Zigangirov
Institute for Information Transmission Problems, RAS, Moscow
Received September 29, 2005; in ﬁnal form, November 14, 2005
Abstract—A lower bound for the number of iteratively correctable erasures is given, with ap-
plication to the ensemble of LDPC codes with parity-check matrices composed of permutation
matrices . We assume that the Zyablov–Pinsker iterative decoding algorithm  is used. Its
complexity is O(N log N), where N is the block length.
In this paper, the erasure correction capability of low-density parity-check (LDPC) codes ,
also called low-density codes, in a binary erasure channel (BEC) is investigated. Unlike most re-
searchers, who studied symbolwise decoding of LDPC codes (see, for example, ), we are interested
in decoding blocks “as a whole.” In general, our paper follows the approach of V.V. Zyablov and
M.S. Pinsker. They studied decoding “as a whole” in both the BEC and the binary symmetric
channel (BSC) [2,5]. In these papers, lower bounds for the number of erasures and errors correctable
by iterative decoding algorithms of LDPC codes in a BEC and BSC were derived.
Particularly, it was shown in  that, if the number of erasures t in a BEC is less than αN ,where
N is the block length and α is a positive constant, then there exists an LDPC code and an iterative
decoding algorithm for this code having complexity O(N log N ) and correcting all conﬁgurations
of t erasures. Note that all iterative erasure-correcting algorithms for LDPC codes that appeared
after the publication of  in 1974 can be reduced to the Zyablov–Pinsker algorithm. The proof
in  was given for an ensemble of binary LDPC codes with a ﬁxed number of ones in each column
of the parity-check matrix but, in contrast to Gallager codes , with variable number of ones in
each row of the parity-check matrix. (Thus, the authors have anticipated the subsequent invention
of irregular LDPC codes.)
A lower bound for the minimum distance of LDPC codes in question, as well as a lower bound
for the number of iteratively corrected erasures, is derived in . This bound is stronger than the
lower bound for the minimum distance of the Gallager codes .
As follows from the Zyablov–Pinsker bound for the number of iteratively correctable erasures
for an LDPC code in a BEC, the probability of iterative decoding failure goes to zero exponentially
with the block length N if the probability of symbol erasure in the BEC is less than α.Notethat,
in the case of symbolwise iterative decoding of LDPC codes in a BEC, an upper bound for the
probability of decoding failure for a block “as a whole” goes to zero as O(N
(see, for example, ).
In this paper, iterative decoding of LDPC codes over a BEC is investigated in the case where
parity-check matrices are constructed from permutation matrices . Our upper bound for the num-