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The Eurasia Proceedings of Science Technology Engineering and Mathematics

On LDPC Codes and Their Construction

Mohammed Amine Tehami, Malika Tehami, Boucif Beddad, Ahmed Hadji

Published Jan 1, 2026• 0 citations
Publisher

Abstract

In this paper, we propose a novel method for constructing low-density parity-check (LDPC) codes that relies on circulant permutation matrices as the main principal elements. A key feature of this construction is the intentional avoidance of short cycles, particularly those of length four, which are known to degrade decoding performance. By eliminating these cycles, the approach contributes to lowering both encoding and decoding complexity, making the codes more efficient to implement. To assess their effectiveness, the proposed LDPC codes are evaluated and compared with uncoded binary phase shift keying (BPSK) transmission over an additive white Gaussian noise (AWGN) channel. The simulation outcomes demonstrate that the structured codes deliver competitive results, achieving error-rate performance similar to, and in some instances exceeding, well-known designs such as progressive edge growth (PEG) and quasi-cyclic (QC) codes.

DOI
10.55549/epstem.1212

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In this paper, we propose a novel method for constructing low-density parity-check (LDPC) codes that relies on circulant permutation matrices as the main principal elements. A key feature of this construction is the intentional avoidance of short cycles, particularly those of length four, which are known to degrade decoding performance. By eliminating these cycles, the approach contributes to lowering both encoding and decoding complexity, making the codes more efficient to implement. To assess their effectiveness, the proposed LDPC codes are evaluated and compared with uncoded binary phase shift keying (BPSK) transmission over an additive white Gaussian noise (AWGN) channel. The simulation outcomes demonstrate that the structured codes deliver competitive results, achieving error-rate performance similar to, and in some instances exceeding, well-known designs such as progressive edge growth (PEG) and quasi-cyclic (QC) codes.
Low-Density Parity-Check (LDPC) codes are among the most effective modern error-correcting codes due to their excellent correction performance and highly parallel decoding scheme. Moreover, the nonbinary extension of such codes further increases performance in the short-block regime. In this paper, we review the key elements for the construction of implementation-oriented binary and nonbinary codes. These Quasi-Cyclic LDPC (QC-LDPC) codes additionally feature efficient encoder and decoder implementation frameworks. We then present a versatile algorithm for the construction of both binary and nonbinary QC-LDPC codes that have low encoding complexity and an optimized corresponding graph structure. Our algorithm uses a progressive edge growth algorithm, modified for QC-LDPC graph construction, and then performs an iterative global search for optimized cyclic shift values within the QC-LDPC circulants. Strong error correction performance is achieved by minimizing the number of short cycles, and cycles with low external connectivity, within the code graph. We validate this approach via error rate simulations of a transmission system model featuring an LDPC coder-decoder, digital modulation, and additive white Gaussian noise channels. The obtained numerical results validate the effectiveness of the proposed construction algorithm, with a number of constructed codes exhibiting either similar or superior performance to industry standard binary codes and selected nonbinary codes from the literature.
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.
In this paper, we propose a novel method with low complexity for constructing quasi-cyclic LDPC codes. The algorithm chooses every circulant permutation matrix sequentially, and makes sure that the current circulant permutation matrix forms no cycles of length 4 and 6 with the existent matrices, finally a family of (3,L)(3,L)(3,L) QC-LDPC codes with girth at least 8 is obtained. The main complexity of the algorithm is checking cycles of length 6, the number of which is no more than L2L^2L2. Simulation results show that, when block length is 1008 and using the iterative belief propagation decoding, the LDPC code constructed by the proposed algorithm out performs the random code about 0.1dB at the bit-error-rate of $ 10^{ - 5}$.
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.
This paper have developed farther the method of constructing sparse parity-check matrix (H-matrix) with determinate systemic structure as well as designing practice recursive encoding algorithm of the low-density parity-check (LDPC) codes. It discovers that the LDPC codes based on the circulant-shifted permutation Q-matrix created by N-queens searching algorithm perform better than them based on permutation identity matrix when Hd-matrix in H =[ HpHd] was constructed by using circulant-shifted permutation Q-matrix and identity matrix, respectively. It presents a simple and efficient algorithm of constructing Hdmatrix without 4-cycle and 6-cycle. The presented LDPC encoders require O(M) operation (M is check-bit size). They are rate-compatible and length- compatible encoders that can cover a very wide range of rate from 0.1 to 0.9 and length including any positive integer from 20 to infinity except prime number and multiple prime numbers. The presented LDPC codes with moderate length and 1/2 rate can achieve reliable performance - 10-5~10-6bit error rate (BER) - on an additive white Gaussian noise (AWGN) channel with signal-to-noise ratio (SNR) Eb/Noabout 1.42 dB1.
Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes have practical significance in the field of engineering, thus their construction algorithm is a hot issue in the field of LDPC codes. Based on existing construction methods, particularly the method based on Progressive Edge-Growth (PEG) algorithm, a new construction method of shift matrix is proposed. It can effectively avoid the copy of cycle structures, especially short cycles in base matrix after extension. Besides, it also relieves the time cost caused by random search, and greatly improves the length of cycles when compared with algorithms based on methods such as quadratic congruence and arithmetic sequence. Those algorithms can only eliminate short cycles with length 4, while our method can provide QC-LDPC codes without length 6 at least. Cycle analysis and simulation results show that the newly constructed QC-LDPC codes have good cycle structure and error correction performance.