ISSN 0032-9460, Problems of Information Transmission, 2012, Vol. 48, No. 4, pp. 297–323.
Pleiades Publishing, Inc., 2012.
Original Russian Text
V.V. Zyablov, P.S. Rybin, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 4, pp. 3–29.
Analysis of the Relation between Properties
of LDPC Codes and the Tanner Graph
V. V. Zyablov and P. S. Rybin
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow
Received October 27, 2011; in ﬁnal form, July 12, 2012
Abstract—A new method for estimating the number of errors guaranteed to be corrected by
a low-density parity-check code is proposed. The method is obtained by analyzing edges with
special properties of an appropriate Tanner graph. In this paper we consider binary LDPC codes
with constituent single-parity-check and Hamming codes and an iterative decoding algorithm.
Numerical results obtained for the proposed lower bound exceed similar results for the best
previously known lower bounds.
Error-correcting capabilities of Gallager’s low-density parity-check (LDPC) codes  for a binary
symmetric channel (BSC) were studied in , where it was shown that there exists Gallager’s LDPC
code capable of correcting a linear portion of errors, with decoding complexity O(n log n), where n is
the LDPC code length. Then in  a new (easier-to-compute) analytical lower bound on the error-
correcting capabilities of the decoder of Gallager’s LDPC code was obtained by using combinatorial
methods, but numerical results in most cases did not exceed those obtained by an old lower bound
from . It should be noted that the decoding algorithm considered in  diﬀers from the algorithm
described in . In this paper we consider an algorithm similar to that from .
An LDPC code with a constituent Hamming code (H-LDPC code) was considered in . The
code distance and “soft” decoding of H-LDPC codes were then investigated in [5, 6]. It was shown
in  that the ensemble of H-LDPC codes contains such codes that the minimum code distance
almost reaches the Gilbert–Varshamov bound. Then a result similar to that from  was ﬁrst
obtained for an H-LDPC code in  by generalization of methods developed in . After that,
using generalized methods from , similar results were obtained in  for a q-ary LDPC code used
over a q-ary symmetric channel, and in  for generalized LDPC codes used over a binary erasure
channel. The authors of  obtained a novel lower bound on the fraction of guaranteed corrected
errors for binary codes on graphs used over a BSC. Numerical results for the lower bound  for
H-LDPC codes exceed similar results for the best previously known lower bounds on error-correcting
capabilities of H-LDPC codes.
In this paper we consider the same Gallager’s LDPC codes and H-LDPC codes and iterative
low-complex hard-decision decoding algorithms as in [2, 11]. We obtain a new lower bound on
the fraction of guaranteed corrected errors by using methods developed in  and generalized
in . The main diﬀerence is in our approach. We consider edges with special properties of the
corresponding Tanner graph instead of the number of constituent codes with unsatisﬁed checks
as was done in [2, 11]. Numerical computation for various choices of LDPC codes shows that the