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IEEE Communications Letters

Large Girth Column-Weight Two and Three LDPC Codes

Mohammad Gholami, Ghaffar Raeisi

Published Aug 21, 2014• 8 citations
Publisher

Abstract

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.

DOI
10.1109/lcomm.2014.2349982

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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 $t$, $3\leq t \leq 20$, $t\neq 7, 15, 16$, 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).

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Non-binary (NB) low-density parity-check (LDPC) codes offer superior error-correcting performance compared to binary LDPC codes, especially for short and medium lengths. Constructing such codes with large girth remains challenging due to various combinatorial constraints. We present a novel algebraic and graph-theoretic construction of non-binary quasi-cyclic (QC) LDPC codes with column weight two and girth 12, which is known to be largest achievable girth for array-based QC-LDPC codes. In constructing girth-12 (2, dc) NB-QC-LDPC codes, our method attains the minimum possible code length for dc < 10. Motivated by the fact that non-binary spatially coupled (SC) QC-LDPC codes achieve optimal performance at column weight three, we further construct NB-SC-QC-LDPC codes with girth ≥ 8 while effectively reducing the number of 8-cycles. Specifically, for (3, dc) codes, our design eliminates dominant absorbing sets such as the (3, 3) and (4, 2), eliminates the (5, 1) elementary absorbing sets (EAS), and eliminates almost all (6, 2) EAS, thereby directly enhancing the error-correction performance. The proposed NB-QC-LDPC codes achieve performance comparable to NB-LDPC codes used in the BeiDou satellite navigation system, while requiring a significantly smaller parity-check matrix storage footprint, for instance, around a 44× reduction in storage requirements with respect to BeiDou rate-1/2 (88, 44) code. Also, the proposed rate-1/2 code with length n = 684 provide a 0.4 dB gain over non-binary near-regular QC-LDPC code with same parameters proposed by Huang et al. [IEEE Trans. Commun.] while requiring a smaller parity-check matrix storage footprint. Furthermore, the proposed rate-1/2 NB-SC-QC-LDPC code having length n = 4608 achieve 0.1 dB coding gain compared to existing protograph constructions while requiring a significantly smaller parity-check matrix storage footprint, i.e., around a 192× reduction.

•Oct 15, 2025•0 citations
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 ttt, 3≤t≤203\leq t \leq 203≤t≤20, t≠7,15,16t\neq 7, 15, 16t=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.
In this paper, some well-structured block designs are presented whose incidence matrices can be considered as the mother matrices of some regular and irregular column-weight three quasi cyclic low-density parity-check codes of girth 20. The class of LDPC codes derived from our construction possesses much more flexibility in terms of code length and rate. Moreover, the length of the regular constructed codes with rate at most 0.5 is at least 4.9 times smaller than the length of the voltage graph LDPC codes with the same rate and girth. Simulation results show that the constructed codes outperform progressive edge growth, repeat accumulate and random LDPC codes.
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.
For a high-rate case, it is difficult to randomly construct good low-density parity-check (LDPC) codes of short and moderate lengths because their Tanner graphs are prone to have short cycles. Also, the existing high-rate quasi-cyclic (QC) LDPC codes can be constructed only for very restricted code parameters. In this paper, based on special classes of cyclic difference families, we propose a new construction method of high-rate regular QC LDPC codes having parity-check matrices consisting of a single row of circulants with column-weight 3 or 4. The proposed QC LDPC codes can be constructed for various code rates and lengths including the minimum achievable length for given column-weight and design rate under girth 6. It is observed that the parity-check matrices of the proposed QC LDPC codes have full rank for column-weight 3 and just one redundant row for column-weight 4. It is shown that the error correcting performance of the proposed QC LDPC codes of short and moderate lengths is almost the same as that of the existing ones through numerical analysis.
Non-binary (NB) low-density parity-check (LDPC) codes offer superior error-correcting performance compared to binary LDPC codes, especially for short and medium lengths. Constructing such codes with large girth remains challenging due to various combinatorial constraints. We present a novel algebraic and graph-theoretic construction of non-binary quasi-cyclic (QC) LDPC codes with column weight two and girth 12, which is known to be largest achievable girth for array-based QC-LDPC codes. In constructing girth-12 (2, dc) NB-QC-LDPC codes, our method attains the minimum possible code length for dc < 10. Motivated by the fact that non-binary spatially coupled (SC) QC-LDPC codes achieve optimal performance at column weight three, we further construct NB-SC-QC-LDPC codes with girth ≥ 8 while effectively reducing the number of 8-cycles. Specifically, for (3, dc) codes, our design eliminates dominant absorbing sets such as the (3, 3) and (4, 2), eliminates the (5, 1) elementary absorbing sets (EAS), and eliminates almost all (6, 2) EAS, thereby directly enhancing the error-correction performance. The proposed NB-QC-LDPC codes achieve performance comparable to NB-LDPC codes used in the BeiDou satellite navigation system, while requiring a significantly smaller parity-check matrix storage footprint, for instance, around a 44× reduction in storage requirements with respect to BeiDou rate-1/2 (88, 44) code. Also, the proposed rate-1/2 code with length n = 684 provide a 0.4 dB gain over non-binary near-regular QC-LDPC code with same parameters proposed by Huang et al. [IEEE Trans. Commun.] while requiring a smaller parity-check matrix storage footprint. Furthermore, the proposed rate-1/2 NB-SC-QC-LDPC code having length n = 4608 achieve 0.1 dB coding gain compared to existing protograph constructions while requiring a significantly smaller parity-check matrix storage footprint, i.e., around a 192× reduction.