ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 3, pp. 205–216.
Pleiades Publishing, Inc., 2015.
Original Russian Text
P.S. Rybin, V.V. Zyablov, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 3, pp. 3–14.
Asymptotic Bounds on the Decoding Error Probability
for Two Ensembles of LDPC Codes
P. S. Rybin and V. V. Zyablov
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow, Russia
e-mail: email@example.com, firstname.lastname@example.org
Received March 16, 2015; in ﬁnal form, June 23, 2015
Abstract—Two ensembles of low-density parity-check (LDPC) codes with low-complexity
decoding algorithms are considered. The ﬁrst ensemble consists of generalized LDPC codes,
and the second consists of concatenated codes with an outer LDPC code. Error exponent
lower bounds for these ensembles under the corresponding low-complexity decoding algorithms
are compared. A modiﬁcation of the decoding algorithm of a generalized LDPC code with a
special construction is proposed. The error exponent lower bound for the modiﬁed decoding
algorithm is obtained. Finally, numerical results for the considered error exponent lower bounds
are presented and analyzed.
A lower bound on the error exponent (error exponent of the total probability of decoding denial
and erroneous decoding), known as Forney’s error exponent, for concatenated codes over a binary
symmetric channel (BSC) was ﬁrst obtained in . Then a similar lower bound, known as the
Blokh–Zyablov bound, was obtained for generalized concatenated codes in . It should be noted
that the decoding complexity of those code constructions is of the order O(n
), where n is the code
Low-density parity-check (LDPC) codes  are known to have the minimal decoding complexity
growth with the code length. In  it was ﬁrst shown that in the LDPC code ensemble there
exist codes capable of correcting a linear portion of errors under a bit-ﬂipping algorithm with
complexity of the order O(n log n). Then in  the method developed in  was modiﬁed for the
case of generalized LDPC codes. In  the estimates from [4, 5] were improved, and in  an
estimate for an irregular LDPC code was obtained. In the present paper we use both the estimate
from  and a slightly modiﬁed estimate from .
The error exponents of expander codes were investigated in [8,9]. It was shown that in this case
there exist codes that attain the capacity of a BSC with positive exponent of error probability under
a low-complexity iterative decoding algorithm. In  a special code construction in the class of
generalized LDPC codes and a low-complexity decoding algorithm were proposed. A lower bound
on the error exponent for these codes under the proposed low-complexity decoding algorithm was
obtained. It was shown for the ﬁrst time that an LDPC code with a special construction exists
such that the error probability of the low-complexity decoding algorithm exponentially decreases
for all code rates below the channel capacity.
The research was carried out at the Institute for Information Transmission Problems of the Russian
Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50-00150.