In this letter, an approach is proposed to increase the row (column)-weight of the parity-check matrix of a 4-cycle free LDPC code such that the constructed LDPC code has girth of at least 6. In fact, to each parity-check matrix H, a new graph Gr(H) (Gc(H)) is assigned in which the vertices correspond to the rows (columns) of H and two vertices are adjacent if and only if the associated rows (columns) have in common at least one nonzero element. Now, in a proper vertex coloring of Gr(H) (Gc(H)), each color is considered as a new column (row) whose nonzero elements happen in the rows (columns) in which the corresponding vertices have the same color. Based on this method, some high-rate LDPC codes with girth 6 and column-weight of at least 4 can be constructed from the recently proposed column-weight three LDPC codes with girth 6. Moreover, using the mutually orthogonal Latin squares, this approach is applied on the incidence matrices of some complete bipartite graphs to generate some girth-6 LDPC codes with different column-weights and large minimum-distances.
In this letter, an approach is proposed to increase the row (column)-weight of the parity-check matrix of a 4-cycle free LDPC code such that the constructed LDPC code has girth of at least 6. In fact, to each parity-check matrix H, a new graph Gr(H) (Gc(H)) is assigned in which the vertices correspond to the rows (columns) of H and two vertices are adjacent if and only if the associated rows (columns) have in common at least one nonzero element. Now, in a proper vertex coloring of Gr(H) (Gc(H)), each color is considered as a new column (row) whose nonzero elements happen in the rows (columns) in which the corresponding vertices have the same color. Based on this method, some high-rate LDPC codes with girth 6 and column-weight of at least 4 can be constructed from the recently proposed column-weight three LDPC codes with girth 6. Moreover, using the mutually orthogonal Latin squares, this approach is applied on the incidence matrices of some complete bipartite graphs to generate some girth-6 LDPC codes with different column-weights and large minimum-distances.
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.
LDPC codes of column weight of two are constructed from minimal distance graphs or cages. Distance graphs are used to represent LDPC code matrices such that graph vertices that represent rows and edges are columns. The conversion of a distance graph into matrix form produces an adjacency matrix with column weight of two and girth double that of the graph. The number of 1's in each row (row weight) is equal to the degree of the corresponding vertex. By constructing graphs with different vertex degrees, we can vary the rate of corresponding LDPC code matrices. Cage graphs are used as examples of distance graphs to design codes with different girths and rates. Performance of obtained codes depends on girth and structure of the corresponding distance graphs.
In this paper, the concept of broken diagonal pair in the chess-like square board is used to define some well-structured block designs whose incidence matrices can be considered as the parity-check matrices of some high rate cycle codes with girth 12. Interestingly, the constructed regular cycle codes with row-weights t, 3 ≤ t ≤ 20, t ≠ 7, 15, 16, have the minimum lengths among the known regular girth-12 cycle codes. In addition, the proposed cycle codes can be easily extended to some high rate column weight-3 LDPC codes with girth 6. Simulation results show that the constructed non-binary cycle codes and binary QC LDPC codes have good performances over AWGN channel.
In this paper, the concept of the {\it broken diagonal pair} in the chess-like square board is used to define some well-structured block designs whose incidence matrices can be considered as the parity-check matrices of some high rate cycle codes with girth 12. The structure of the proposed parity-check matrices significantly reduces the complexity of encoding and decoding. Interestingly, the constructed regular cycle codes with row-weights t, 3≤t≤20, t=7,15,16, have the best lengths among the known regular girth-12 cycle codes. In addition, the proposed cycle codes can be easily extended to some high rate column weight-3 LDPC codes with girth 6. Simulation results show that the constructed codes achieve excellent performances, specially the constructed column weight 3 LDPC codes outperform LDPC codes based on Steiner triple systems (STS).
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.
This article presents a highly flexible method for constructing high-girth quasi-cyclic low-density parity-check (QC-LDPC) codes. The proposed algorithm constructs a Tanner graph formed by connecting groups of rows (check nodes) and columns (variable nodes) of the constructed code. To obtain a sub-matrix structure, rows and columns are divided into groups of equal sizes. Rows and columns in a group are connected in their numerical (positional) order to obtain a cyclic structure. Connected rows and columns must satisfy the desired minimum cycle length. We present conditions that guarantee desired girths. The proposed algorithm is by far more flexible in constructing a wide range (rates and lengths) of regular and irregular QC-LDPC codes compared to existing methods. The algorithm, which has linear complexity with respect to the number of rows or columns, provides an easy and fast way to construct QC-LDPC codes. Constructed codes show good bit error rate performances.