ISSN 0032-9460, Problems of Information Transmission, 2013, Vol. 49, No. 4, pp. 333–347.
Pleiades Publishing, Inc., 2013.
Original Russian Text
F.I. Ivanov, V.V. Zyablov, 2013, published in Problemy Peredachi Informatsii, 2013, Vol. 49, No. 4, pp. 41–56.
Low-Density Parity-Check Codes Based on Steiner
Systems and Permutation Matrices
F. I. Ivanov and V. V. Zyablov
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow
Received May 15, 2013; in ﬁnal form, August 27, 2013
Abstract—An algorithm for generating parity-check matrices of regular low-density parity-
check codes based on permutation matrices and Steiner triple systems S(v, 3, 2), v =2
is proposed. Estimations of the rate, minimum distance, and girth for obtained code construc-
tions are presented. Results of simulation of the obtained code constructions for an iterative
“belief propagation” (Sum-Product) decoding algorithm applied in the case of transmission over
a binary channel with additive Gaussian white noise and BPSK modulation are presented.
Low-density parity-check codes (LDPC codes) were proposed by Gallager in . These are
linear block codes deﬁned by their parity-check matrices H characterized by a relatively small
number of ones in their rows and columns. It is often convenient to consider an LDPC code as its
Tanner graph , where connected symbol and code vertices are used for representation of rows
and columns of a parity-check matrix.
An important characteristic of an LDPC code is absence of cycles of certain lengths. A cycle
of length 4 (4-cycle) can be understood as a rectangle in the parity-check matrix whose vertices
are ones. The absence of 4-cycles can be deﬁned with the help of scalar product of all rows (or
columns) in the parity-check matrix. If every pairwise scalar product of rows (or columns) in the
parity-check matrix is not greater than 1, then 4-cycles are absent. Cycles of larger lengths are
deﬁned by the girth of the Tanner graph.
Apart from random LDPC codes, various algebraic constructions of low-density parity-check
codes based on permutation matrices [3–12], projective geometries , and other combinatorial
constructions [14, 15] are often used in practice. The main advantage of such approach is the
possibility to obtain code construction with deterministic characteristics such as girth and minimum
The main objective of this work is to construct and explore properties of an ensemble of low-
density parity-check codes based on two algebraic constructions simultaneously: Steiner triple
systems S(v,3, 2) and permutation matrices. The authors are not aware of works where Steiner
triple systems and permutation matrices are simultaneously used for constructing parity-check
matrices of LDPC codes.
We propose a simple generation method for parity-check matrices of such codes with v =2
For the obtained ensemble, we give lower and upper bounds on the code rate and lower bounds
on the minimum distance and girth bound were. For the proposed constructions of parity-check
matrices, we prove that the girth is at least 6 and the minimum distance is at least 4.