ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 2, pp. 142–159.
Pleiades Publishing, Inc., 2010.
Original Russian Text
A.A. Frolov, V.V. Zyablov, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 2, pp. 47–65.
Asymptotic Estimation of the Fraction of Errors
Correctable by q-ary LDPC Codes
A. A. Frolov and V. V. Zyablov
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received November 23, 2009; in ﬁnal form, February 5, 2010
Abstract—We consider an ensemble of random q-ary LDPC codes. As constituent codes, we
use q-ary single-parity-check codes with d = 2 and Reed–Solomon codes with d =3. Wepropose
a hard-decision iterative decoding algorithm with the number of iterations of the order of the
logarithm of the code length. We show that under this decoding algorithm there are codes in
the ensemble with the number of correctable errors linearly growing with the code length. We
weaken a condition on the vertex expansion of the Tanner graph corresponding to the code.
Low-density parity-check (LDPC) codes were proposed by Gallager in . They are characterized
by sparse parity-check matrices. If a parity-check matrix contains j ones in every column and k ones
in every row, the code is said to be a regular (j, k) LDPC code. Graphically, the code can be
represented as a bipartite graph with symbol vertices corresponding to columns of the parity-check
matrix (they are of degree j) and code vertices corresponding to rows and having degree k.This
method was proposed by Tanner in . An example of a bipartite Tanner graph is shown in Fig. 1.
Gallager’s LDPC code construction is a particular case of the Tanner construction and is ob-
tained from it when using a binary single-parity-check code as a constituent code. Alternative
constructions of LDPC codes can be obtained by replacing single-parity-check codes with other
block codes of lengths equal to degrees of the vertices. As constituent codes, one can use Ham-
ming [3, 4], BCH, or Reed–Solomon  codes.
The error-correcting capability of Gallager’s LDPC codes for a binary symmetric channel (BSC)
was ﬁrst considered in , where existence of LDPC codes capable of correcting a linearly growing
(with code length) number of errors with decoding complexity O(n log
n) was proved, where n
is the code length. Hereafter, by the complexity we mean the deﬁnition given in , i.e., the
minimum number of functional elements in a circuit realizing the decoding. There are similar
results for LDPC codes with constituent Hamming codes .
In the present paper, developing the ideas of [3,6], we prove that among random q-ary codes with
aconstituentq-ary single-parity-check code or a constituent Reed–Solomon code with minimum
code distance d = 3 there are codes with small decoding complexity (O(n log
n)) that are capable
of correcting a linearly growing (with code length) number of errors. Thus, we generalize the results
obtained in  for binary codes with d = 2 and in  for binary codes with d = 3 to the case of
q-ary codes. This is our main goal.
Besides, in the present paper we propose a generalization of the algorithm used in  to the case
of q-ary codes.