We give a construction of LDPC codes from Deza graphs with parameters
(v,k,1,0) . The Tanner graphs of these LDPC codes do not contain cycles of length 4. Special attention is given to the construction of LDPC codes from the Moore graphs with diameter 2. We identify and enumerate the smallest absorbing sets in the Tanner graphs of the obtained LDPC codes and analyze their structures. Furthermore, we describe a construction, based on the protograph operation, of an infinite family of LDPC codes, whose Tanner graphs have girth at least 6, obtained from each
(v,k,1,0) Deza graph. We give an expression for the variance of a syndrome weight of the constructed LDPC codes, and also present simulation results.
In this paper, we study binary low-density parity-check, or LDPC codes, determined by the adjacency matrices of the odd graphs as their parity-check matrices. We show that for the odd graph $ O_n $, $ n\geq 3 $, the obtained code $ C_n $ is an $ (n, n) $-regular binary LDPC code of length $ {2n-1 \choose n-1} $, dimension $ {2n-2 \choose n-2} $ and minimum distance $ n+1 $. Further, we show that the girth of the corresponding Tanner graph determined by the adjacency matrix of an odd graph is equal to 6, and give the expression for the variance of the syndrome weight for $ C_n $. Moreover, we show that there are no absorbing sets of size smaller than 3 for these codes and that the absorbing sets of size 3 exist only for $ n = 3 $. Additionally, we give the structure of $ (a, b) $ absorbing sets with $ b \leq a\leq n $. Finally we prove that $ C_n $ is also a linear code with complementary dual, or LCD code, for $ n\geq3 $.
The main subjects of the thesis are LDPC codes constructed from combinatorial structures. Combinatorial structures that have been used for the construction are:
μ-geodetic graphs obtained from block designs, strongly regular graphs with parameters
(v,k,0,1), and block designs with parameters
(45,5,1). In the thesis we have constructed a family of LDPC codes using
μ-geodetic graphs obtained from block
(v,k,λ) designs. We have shown that the Tanner graphs of LDPC codes constructed from
μ-geodetic graphs obtained from block designs with
k=3 are free of cycles of length four, and we have analyzed properties of the constructed codes. We have examined a connection between the parameters of the constructed codes and the parameters of the initial design. The existence and the structures of absorbing sets in the Tanner graphs of the constructed codes have been analyzed. Analysis of structures of the absorbing sets enabled us to determine the exact number of the absorbing sets of certain size using parameters of the initial design. For construction of LDPC codes we have used strongly regular graphs with parameters
(v,k,0,1) since the Tanner graph of the corresponding LDPC code is free of cycles of length four. We have examined the existence and the structures of absorbing sets in the Tanner graphs of the constructed codes. LDPC codes from some block designs with parameters
(45,5,1) have been constructed. Simulation results for the constructed LDPC codes have been presented.
Absorbing sets, a class of combinatorial structures of the Tanner graph representation of a low-density parity-check (LDPC) code, are known to influence the performance of the code under message passing iterative decoding. In this paper, we study the smallest absorbing sets of LDPC codes constructed from projective planes and Euclidean planes. The lower bounds on the parameters of smallest absorbing sets given by Dolecek are proven to be tight for these two families of LDPC codes. We also analyze the combinatorial properties of the smallest absorbing sets and give conditions necessary and sufficient for a set of bit nodes in the Tanner graph to be a smallest absorbing set. For LDPC codes from projective planes, we further give a condition necessary and sufficient for a smallest absorbing set to be a fully absorbing set. In addition, we show that these smallest absorbing sets are asymptotically not stable, which may explain to some extent the good performance as well as the low error floor expectation of these two families of LDPC codes.
In contrast to the capacity approaching performance of iteratively decoded low-density parity check (LDPC) codes, many practical finite-length LDPC codes exhibit performance degradation, manifested in a so-called error floor. Previous work has linked this phenomenon to the presence of certain combinatorial structures within the Tanner graph representation of the code, termed absorbing sets. Absorbing sets are stable under the bit-flipping operations and have been shown to act as fixed points (¿absorbers¿) for a wider class of iterative decoding algorithms. Codes often possess absorbing sets whose size is smaller than the minimum distance: the smallest absorbing sets are deemed most detrimental culprits behind the error floor. This paper focuses on the elementary combinatorial bounds of the smallest (candidate) absorbing sets. For certain classes of practical codes we demonstrate the tightness of these bounds and show how can the structure of the code and the structure of the absorbing sets be utilized to increase the size of the smallest absorbing sets without compromising other code properties such as the node degrees and the girth. As such, this work provides a step towards a better code design by taking into account the combinatorial nature of fixed points of iterative decoding algorithms.