ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 2, pp. 95–109.
Pleiades Publishing, Inc., 2009.
Original Russian Text
V.V. Zyablov, R. Johannesson, M. Lonˇcar, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 2, pp. 25–40.
Low-Complexity Error Correction of
Hamming-Code-Based LDPC Codes
V. A. Zyablov
, R. Johannesson
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Dept. of Electrical and Information Technology (EIT), Lund University, Sweden
Nokia, Devices R&D, Modem Algorithm Design, Copenhagen, Denmark
Received May 3, 2008; in ﬁnal form, February 24, 2009
Abstract—Ensembles of binary random LDPC block codes constructed using Hamming codes
as constituent codes are studied for communicating over the binary symmetric channel. These
ensembles are known to contain codes that asymptotically almost meet the Gilbert–Varshamov
bound. It is shown that in these ensembles there exist codes which can correct a number of
errors that grows linearly with the code length, when decoded with a low-complexity iterative
decoder, which requires a number of iterations that is a logarithmic function of the code length.
The results are supported by numerical examples, for various choices of the code parameters.
Low-Density Parity-Check (LDPC) codes with constituent single-parity-check codes, invented
by Gallager , are characterized by sparse parity-check matrices. If the matrix contains j ones
in each column and k ones in each row, the code is referred to as a (j, k)-regular LDPC code.
In the bipartite graph associated with such a code , all variable nodes, which represent code
symbols, correspond to all columns of the parity-check matrix and have degree j, and all constraint
nodes correspond to all rows of the parity-check matrix and have degree k. Each constraint node
represents a single-parity-check (SPC) code over k variable nodes connected to it. The error-
correcting capabilities of random Gallager’s LDPC codes when communicating over the binary
symmetric channel (BSC) were studied in , where it was shown that there exist LDPC codes
capable of correcting a portion of errors that grows linearly with the code length n, with decoding
complexity O(n log n). Note that, hereinafter, our deﬁnition of complexity follows the one given
in ; i.e., it is the minimum required number of functional elements in a scheme that realizes
Alternative constructions of LDPC codes can be obtained by replacing the SPC codes associated
with the graph’s constraint nodes with other constituent block codes of length equal to the check-
node degree. For example, Hamming codes [5–7], Reed–Solomon codes , or BCH codes  can
be used as constituent codes. The so-obtained codes are often referred to as generalized LDPC
codes. These LDPC codes can all be regarded as special cases of hypergraph-based codes .
To obtain an ensemble of hypergraph codes with a constituent code C
of length n
from an -partite, n
-regular, -uniform hypergraph H =(V, E). The set V of the hypergraph
Supported in part by the Royal Swedish Academy of Sciences in cooperation with the Russian Academy
of Sciences and by the Swedish Research Council, Grant no. 621-2007-6281.