ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 4, pp. 357–377.
Pleiades Publishing, Inc., 2009.
Original Russian Text
I.E. Bocharova, B.D. Kudryashov, R.V. Satyukov, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 4,
Graph-Based Convolutional and Block LDPC Codes
I. E. Bocharova
, and R. V. Satyukov
St. Petersburg State University of Information Technologies, Mechanics and Optics
firstname.lastname@example.org email@example.com firstname.lastname@example.org
Received June 9, 2009; in ﬁnal form, September 14, 2009
Abstract—We consider regular block and convolutional LDPC codes determined by parity-
check matrices with rows of a ﬁxed weight and columns of weight 2. Such codes can be described
by graphs, and the minimum distance of a code coincides with the girth of the corresponding
graph. We consider a description of such codes in the form of tail-biting convolutional codes.
Long codes are constructed from short ones using the “voltage graph” method. On this way
we construct new codes, ﬁnd a compact description for many known optimal codes, and thus
simplify the coding for such codes. We obtain an asymptotic lower bound on the girth of the
corresponding graphs. We also present tables of codes.
In recent years, LDPC (low-density parity-check) codes proposed by Gallager  attract increas-
ing interest as an alternative to turbo codes . The known LDPC codes can be divided into two
groups: irregular (random or pseudorandom) [3,4] and regular [5,6] codes.
Aregular(J, L) LDPC code is determined by a binary parity-check matrix where each row
contains precisely L ones, and each column, precisely J ones. Among regular LDPC codes, the
class of quasi-cyclic codes is distinguished; these are codes such that for some positive integer p
the shift of any codeword by p positions is also a codeword [6, 7]. Quasi-cyclicity plays an im-
portant role in practical use of LDPC codes, since it makes it possible to simplify the coding
procedure. An important parameter of an LDPC code is the girth g
of the corresponding Tan-
ner graph . On the parameter g
, eﬃciency of iterative decoding procedures for LDPC codes
In the present paper we consider a subclass of LDPC codes: those with J = 2. The parity-check
matrix of codes of this class can be considered as an incidence matrix of some graph; therefore,
such codes are also called graph codes. Although it is known (see, e.g., ) that among these codes
there are no asymptotically optimal codes, interest towards them (see [9–11]) is due to their low
maximum-likelihood decoding complexity and to the existence of rather eﬃcient short codes in
this class. Moreover, combined with other codes, they can be used in concatenated constructions
(see, e.g., ).
Note that the minimum distance of a graph code coincides with the girth g of the corresponding
graph and equals g = g
We consider representations of quasi-cyclic LDPC codes in the form of tail-biting convolutional
codes [14, 15]. Such a description was obtained for many (more precisely, for almost all known)
optimal graph codes satisfying the Moore lower bound [16, 17]. Thus, simple coding schemes are
Supported in part by the Royal Swedish Academy of Sciences and the Russian Academy of Sciences, grant