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IEEE/CIC International Conference on Communications in China (ICCC)

Random Constructions of Girth-Eight QC-LDPC Codes with Structural Shaping

Songyan Xue, Guohua Zhang

Published Aug 10, 2025• 0 citations
Publisher

Abstract

In order to construct girth-8 quasi-cyclic (QC) low-density parity-check (LDPC) codes with low encoding complexity, this paper introduces structural shaping techniques to generate QC-LDPC codes with two different structures. These two structures are QC-LDPC codes constructed with vertical symmetric structure and reverse structure without 4-cycles and 6-cycles. Novel constructions assume that the structures of the parity check matrix (PCM) are symmetric and reverse, respectively, and the greedy exhaustive search algorithm is used to find the exponent matrix with the smallest possible size P of the circulant permutation matrix (CPM). Due to its symmetrical and reverse structural characteristics, the representation of the new codes is more compact than other codes. The simulation results show that the proposed codes have certain competitiveness compared to existing QC-LDPC codes.

DOI
10.1109/icccworkshops67136.2025.11148129

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•Jan 1, 2009•1 citation
In order to construct girth-8 quasi-cyclic (QC) low-density parity-check (LDPC) codes with low encoding complexity, this paper introduces structural shaping techniques to generate QC-LDPC codes with two different structures. These two structures are QC-LDPC codes constructed with vertical symmetric structure and reverse structure without 4-cycles and 6-cycles. Novel constructions assume that the structures of the parity check matrix (PCM) are symmetric and reverse, respectively, and the greedy exhaustive search algorithm is used to find the exponent matrix with the smallest possible size P of the circulant permutation matrix (CPM). Due to its symmetrical and reverse structural characteristics, the representation of the new codes is more compact than other codes. The simulation results show that the proposed codes have certain competitiveness compared to existing QC-LDPC codes.
In this paper, we propose new constructions for regular girth-8 quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices (CPM). The constructions assume symmetries in the structure of the parity-check matrix and employ a greedy exhaustive search algorithm to find the permutation shifts of the CPMs. As a result of symmetries, the new codes have a more compact representation compared with their counterparts. In majority of cases, also, they achieve the girth 8 at a shorter block length for the same degree distribution (code rate). Deterministic (explicit) constructions are also presented to expand the proposed parity-check matrices to larger block lengths and higher rates. The proposed long high-rate codes are often substantially shorter than regular girth-8 QC-LDPC codes of similar rate in the literature. Simulation results demonstrate that the proposed symmetric codes have competitive performance in comparison with similar existing QC-LDPC codes that lack symmetry.
By classifying cycle types within the parity-check matrix of a quasi-cyclic(QC) low-density parity-check(LDPC) code,a method without repetition or omission is proposed to detect all the cycles with lengths smaller than twelve.Based on this cycle detection method,a new approach is presented to construct large girth QC-LDPC codes by using some heuristic strategies.The approach includes three steps.Firstly,under the infinite cyclic permutation matrix(CPM) size condition,two integer sequences satisfying the girth constraint are found using a heuristic strategy.Secondly,according to the designed code rate,several integer pairs are extracted from the two sequences so as to form a shift matrix.Finally,by employing the recently proposed theoretical lower bound by Zhang et.al,a range of consecutive CPM sizes in which the girth never decrease is accurately computed from the shift matrix.Compared with Liu-Han's method,the new approach possesses two advantages in the sense that:(1) Not only girth-10+ QC-LDPC codes but also girth-12 ones can be constructed to have consecutive CPM sizes;(2) girth-10+ and girth-12 QC-LDPC codes can be generated with very high code rates,due to the new cycle detection method which significantly accelerates the sequence search procedure.
Enabling the parity-check matrices to have a large girth is an efficient method to improve the decoding performance for many binary and non-binary QC-LDPC codes.A new explicit framework to construct QC-LDPC codes with girth at least eight is proposed,without any computer search procedures.The QC-LDPC codes constructed by the framework not only have a girth at least eight,but also possess an advantage of consecutive cyclic permutation matrix(CPM) sizes.The framework includes two steps:first,a parity-check matrix is explicitly constructed with a girth at least eight under an infinite CPM size;second,the tight lower bound of consecutive CPM sizes is precisely calculated directly from the corresponding shift matrix of the parity-check matrix,by employing a new girth property discovered by this paper.
In this correspondence we present a special class of quasi-cyclic low-density parity-check (QC-LDPC) codes, called block-type LDPC (B-LDPC) codes, which have an efficient encoding algorithm due to the simple structure of their parity-check matrices. Since the parity-check matrix of a QC-LDPC code consists of circulant permutation matrices or the zero matrix, the required memory for storing it can be significantly reduced, as compared with randomly constructed LDPC codes. We show that the girth of a QC-LDPC code is upper-bounded by a certain number which is determined by the positions of circulant permutation matrices. The B-LDPC codes are constructed as irregular QC-LDPC codes with parity-check matrices of an almost lower triangular form so that they have an efficient encoding algorithm, good noise threshold, and low error floor. Their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices.
A novel class of quasi-cyclic (QC) low-density parity-check (LDPC) codes is explicitly proposed without 4-cycles and 6-cycles via the greatest-common-divisor (GCD) framework. From the new codes, a novel type of girth-eight QC-LDPC codes with a dual-diagonal (DD) structure is presented, in which the single undetermined entry in the associated exponent matrix can be determined by simple formulae. The proposed DD codes have similar or better performance compared with DD codes obtained from computer search. Moreover, a lower bound on circulant sizes which guarantees the existence of qualified undetermined entries is also provided.
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}$.