In this study, a method is presented to construct column weight 3 (CW3) low‐density parity‐check (LDPC) codes using three‐partite graphs. Let G b be a bipartite graph and N g be the set of all minimum length cycles in G b . Using G b and N g , a three‐partite graph denoted G ( G b , N g ), or simply G t , is formed. Let T be the set of length 3 cycles in G t and T a be the set of three element subsets of vertices in G t such that each of these subsets form a subgraph with no edges in G t and has precisely one element in each section of G t . Furthermore, let H be the binary matrix in which the set of rows represent the set of vertices of G t , the columns represent the elements of V := T ∪ T a , and h ij = 1 if and only if the i th vertex of G t belongs to the j th three element set in V . Then H is a CW3 binary matrix. Using the Tanner graph representing H , a recursive construction for CW3 LDPC codes is provided. Applying a simple restriction on T and T a , codes free of length 4 cycles are generated. Euclidean and finite geometry codes are used as the base codes for generating new CW3 LDPC codes. Results are presented which show that these new codes perform well in an additive white Gaussian noise (AWGN) channel with the iterative sum‐product decoding algorithm.
In this study, a method is presented to construct column weight 3 (CW3) low‐density parity‐check (LDPC) codes using three‐partite graphs. Let G b be a bipartite graph and N g be the set of all minimum length cycles in G b . Using G b and N g , a three‐partite graph denoted G ( G b , N g ), or simply G t , is formed. Let T be the set of length 3 cycles in G t and T a be the set of three element subsets of vertices in G t such that each of these subsets form a subgraph with no edges in G t and has precisely one element in each section of G t . Furthermore, let H be the binary matrix in which the set of rows represent the set of vertices of G t , the columns represent the elements of V := T ∪ T a , and h ij = 1 if and only if the i th vertex of G t belongs to the j th three element set in V . Then H is a CW3 binary matrix. Using the Tanner graph representing H , a recursive construction for CW3 LDPC codes is provided. Applying a simple restriction on T and T a , codes free of length 4 cycles are generated. Euclidean and finite geometry codes are used as the base codes for generating new CW3 LDPC codes. Results are presented which show that these new codes perform well in an additive white Gaussian noise (AWGN) channel with the iterative sum‐product decoding algorithm.
This paper introduces a class of structured low-density parity-check (LDPC) codes whose parity check matrices are arrays of permutation matrices. The permutation matrices are obtained from Latin squares and form a finite field under some matrix operations. They are chosen so that the Tanner graphs do not contain subgraphs harmful to iterative decoding algorithms. The construction of column-weight-three codes is presented. Although the codes are optimized for the Gallager A/B algorithm over the binary symmetric channel (BSC), their error performance is very good on the additive white Gaussian noise channel (AWGNC) as well.
Abstract A column‐weight LDPC code with the parity‐check matrix is called decomposable if there exists a permutation on the rows of , such that can be decomposed into column‐weight one matrix. In this paper, some variations of edge coloring of graphs are used to construct some column‐weight three decomposable LDPC codes with girths at least six and eight. Applying the presented method on several known classes of bipartite graphs, some classes of column‐weight three decomposable LDPC codes are derived having flexibility in length and rate. Interestingly, the constructed parity‐check matrices based on the proper edge coloring of graphs can be considered as the base matrix of some high rate column‐weight three quasi‐cyclic (QC) LDPC codes with maximum‐achievable girth 20. The paper also leads to a simple characterization of elementary trapping sets of the decomposable codes based on the chromatic index of the corresponding normal graphs. This characterization corresponds to a simple search algorithm finds all possible existing elementary trapping sets in a girth‐6 or girth‐8 column‐weight 3 LDPC code which are layered super set of a short cycle in the Tanner graph of the code. Simulation results indicate that the QC‐LDPC codes with large girths lifted from the constructed base matrices have good performances over AWGN channel.
In this paper a new low complexity method for constructing binary quasi-cyclic low-density parity-check (QC-LDPC) codes is introduced. In the proposed method, each block-column of the parity check matrix H is made by a circulant matrix in a way that the associated Tanner graph is free of cycle four. Each circulant matrix in H is made by a generator column. The generator columns should be selected in a way that each associated circulant matrix and every two distinct circulant matrices are free of cycle four. The generator columns are made by row distance sets. An algorithm for generating distance sets and obtaining circulant matrices with columns of weight three is presented separately. Simplicity of construction and having a good flexible family of quasi cyclic LDPC codes both in rate and length are the main properties of the proposed method. The performance of the proposed codes is compared with that of the random-like and Array LDPC codes over an AWGN channel. Simulation results show that from the performance perspective, the constructed codes are competitive with random-like and Array LDPC codes.
In this paper, a new type of edge coloring of graphs together with an algorithm for such an edge coloring is presented to construct some column-weight three low-density parity-check (LDPC) codes whose Tanner graphs are free of 4-cycles. This kind of edge coloring is applied on some well-known classes of graphs such as complete graphs and complete bipartite graphs to generate some column-weight 3 LDPC codes having flexibility in terms of code length and rate. Interestingly, the constructed (3, k)-regular codes with regularities k = 4, 5,..., 22 have lengths n = 12, 20, 26, 35, 48, 57, 70, 88, 104, 117, 140, 155, 176, 204, 228, 247, 280, 301, 330, having minimum block length compared to the best known similar codes in the literature. In addition to linear complexity of generating such parity-check matrices, they can be considered as the base matrices of some quasi-cyclic (QC) LDPC codes with maximum achievable girth 18, which inherit the low-complexity encoder implementations of QC-LDPC codes. Simulation results show that the QC-LDPC codes with large girth lifted from the constructed base matrices have good performances and outperform random codes, progressive edge growth LDPC codes, some finite fields and group rings based QC-LDPC codes and also have a close competition to the standard IEEE 802. 16e (WiMAX) code.
This paper,based on algebra and graph,proposes a method for constructing structured low-density parity-check(LDPC) codes.In this method,the authors design a connected graph with three kinds of special path to ensure that the Tanner graph of the parity check matrix mapped from the connected graph is without short cycles.The construction method results in two classes of(3,k)-regular quasi-cyclic LDPC codes with girth 8 and 12 respectively.Furthermore,by extending this construction method,two classes of LDPC codes with column weight 2,whose girths are respectively 16 and 24,can be obtained.The simulation studies show that these codes can achieve better performance than randomly constructed regular LDPC codes over AWGN channels with iterative decoding.At the BER of 10-5,the new proposed(3,6)-regular(4356,2205) QC-LDPC code performs 1.48dB from the Shannon limit.Moreover,it outperforms the random regular LDPC code of comparable parameters by 0.1dB.For the new proposed structured LDPC codes with column weight 2,they can achieve about 2dB coding gains with respect to the random regular LDPC codes with similar block lengths and code rates.
Low Density Parity Check (LDPC) codes are specified by the parity check matrix H. Smaller girths in the Tanner graphs of LDPC codes prevent the sum-product algorithm from converging, i.e., short cycles in H-matrix degrade the performance of LDPC decoder. In this paper, we present an algorithm for constructing LDPC codes without short cycles. The H-matrix must be sufficiently sparse to increase the girth of a code. In this work, we present a simple method to increase the sparsity of matrix and provide lemmas to avoid short cycles of lengths 4 and 6. We design sub-matrices for column-weight two and column-weight three codes. This design makes use of a simple shifting function given by the proposed scheme. These are then combined into a final H-matrix using Gray code representations. The proposed method provides structured, regular codes with flexibility in code rates and code lengths. The regularity of these codes gives the advantages like simplicity of hardware implementation and fast encoding. Key words LDPC codes, Tanner Graph, Girth, Gray codes