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Axioms

Construction of Multi-Rate QC-LDPC Codes Based on Permutation Method

Hengzhou Xu, Jinru Wang, Mei Zhang, Mengmeng Xu, Qian Wang

Published Aug 3, 2026• 0 citations
Publisher

Abstract

This paper proposes a systematic permutation-based construction method for multi-rate quasi-cyclic low-density parity-check (QC-LDPC) codes. We first present a graph-theoretic framework in which any regular QC-LDPC code can be normalized to a canonical base matrix that is uniquely determined by a permutation π. This normalization reduces the complex code design to a single combinatorial optimization problem over the symmetric group. Based on this normalization, we analyze the cycle structure of the lifted Tanner graph and derive necessary and sufficient conditions for 4-cycle elimination in terms of the permutation difference function. We develop two complementary algorithms: a simulated annealing algorithm that searches for permutations that minimize a weighted sum of 4-cycles and 6-cycles, and a progressive column-ordering algorithm that ensures every prefix subgraph maintains high girth. This approach yields a nested family of rate-compatible codes. Simulation results show that the constructed codes outperform the 5G-LDPC codes. The nested base matrix structure facilitates seamless rate switching, which makes the proposed code family well suited for adaptive transmission systems in future wireless networks.

DOI
10.3390/axioms15080583

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This paper proposes a systematic permutation-based construction method for multi-rate quasi-cyclic low-density parity-check (QC-LDPC) codes. We first present a graph-theoretic framework in which any regular QC-LDPC code can be normalized to a canonical base matrix that is uniquely determined by a permutation π. This normalization reduces the complex code design to a single combinatorial optimization problem over the symmetric group. Based on this normalization, we analyze the cycle structure of the lifted Tanner graph and derive necessary and sufficient conditions for 4-cycle elimination in terms of the permutation difference function. We develop two complementary algorithms: a simulated annealing algorithm that searches for permutations that minimize a weighted sum of 4-cycles and 6-cycles, and a progressive column-ordering algorithm that ensures every prefix subgraph maintains high girth. This approach yields a nested family of rate-compatible codes. Simulation results show that the constructed codes outperform the 5G-LDPC codes. The nested base matrix structure facilitates seamless rate switching, which makes the proposed code family well suited for adaptive transmission systems in future wireless networks.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.
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.
A new approach is proposed for constructing regular low-density parity-check (LDPC) codes based on tensor product of matrices. In this paper, first a general construction method of regular LDPC codes exploiting permutation matrices is described. Constructed codes have a quasi-cyclic structure with no short cycles of length 4 in their Tanner graph, hence simple encoding while maintaining good performance is achieved. The paper also demonstrates a generalized design, which covers a large family of LDPC codes and number of other construction methods. The new generalized LDPC codes are defined by a small number of parameters and cover a large set of code lengths and rates. Using these codes, LDPC matrices of any column weight and row weight can be constructed. Performance of these codes under iterative decoding compares well with other well-structured as well as random LDPC codes
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.
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 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.