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IEEE Communications Letters

Dual-Diagonal QC-LDPC Codes Free of Small Cycles From GCD Framework via Negative Coefficients

Guohua Zhang, Mengjuan Lou, Aijing Sun, Yi Fang

Published Oct 4, 2024• 5 citations
Publisher

Abstract

A novel class of dual-diagonal (DD) quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight is proposed. For the exponent matrix of such a DD-QC-LDPC code, the left sub-matrix is explicitly generated via new formulae which are motivated by a search algorithm based on the greatest-common-divisor (GCD) framework and a new strategy allowing negative coefficients. The right sub-matrix (DD matrix) is designed by modifying the existing DD matrix while preserving the simple encoding property directly through the parity-check matrix. Performance simulations indicate that the proposed DD-QC-LDPC codes significantly outperform several existing counterparts with or without DD structure. Unlike existing formulae which involve only non-negative coefficients, the new formulae motivated by proposed search algorithm allow for negative coefficients, which lead to significantly improved lower bounds on circulant sizes for column weights from five to seven. As a result, the new DD-QC-LDPC codes not only enable simple encoding and low description complexity, but also possess consecutive and reasonably small circulant sizes, flexible sizes for exponent matrices as well as good decoding performance.

DOI
10.1109/lcomm.2024.3474209

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A novel class of dual-diagonal (DD) quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight is proposed. For the exponent matrix of such a DD-QC-LDPC code, the left sub-matrix is explicitly generated via new formulae which are motivated by a search algorithm based on the greatest-common-divisor (GCD) framework and a new strategy allowing negative coefficients. The right sub-matrix (DD matrix) is designed by modifying the existing DD matrix while preserving the simple encoding property directly through the parity-check matrix. Performance simulations indicate that the proposed DD-QC-LDPC codes significantly outperform several existing counterparts with or without DD structure. Unlike existing formulae which involve only non-negative coefficients, the new formulae motivated by proposed search algorithm allow for negative coefficients, which lead to significantly improved lower bounds on circulant sizes for column weights from five to seven. As a result, the new DD-QC-LDPC codes not only enable simple encoding and low description complexity, but also possess consecutive and reasonably small circulant sizes, flexible sizes for exponent matrices as well as good decoding performance.
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.
For any column weight J and any row weight L, a novel framework is proposed such that a girth-eight (J,L) quasi-cyclic low-density parity-check (QC-LDPC) code with any block length above a lower bound can be constructed via a simple inequality in terms of greatest common divisor (GCD). The main advantage is that the construction of a class of (J,L) girth-eight QC-LDPC codes is transformed into a rather simple task, searching for J integers satisfying the so-called GCD constraint for L. Combining the new method with masking matrices, a class of type-1 QC-LDPC codes is presented with girth at least eight. Simulation results show that the type-1 codes perform better than the random QC-LDPC codes and quadratic-congruence-based QC-LDPC codes for moderate block lengths and low code rates.
A construction method for girth‐eight quasi‐cyclic low‐density parity‐check codes (QC‐LDPCs) with low encoding complexity is proposed in this study. To avoid short cycles and further improve the performance of the authors proposed codes, the greatest common divisor (GCD)‐based method is utilised to construct the information part with girth‐eight of the parity‐check matrix with systematic form. Furthermore, the quasi‐dual‐diagonal structure is adopted as the parity part of the parity‐check matrix for fast encoding, which not only maintains the parity‐check matrix with girth‐eight, but also efficiently decreases the encoding complexity of their proposed codes. Simulation results show that their proposed codes perform better than Mackay's codes and the QC‐LDPCs constructed by the GCD‐based method, and have comparable performance compared with progressive edge growth based QC‐LDPCs. Besides, their proposed codes have lower encoding complexity due to the property of fast encoding.
This paper proposes a new explicit construction method for quasi-cyclic (QC) low-density parity-check (LDPC) codes without 4-cycles and 6-cycles with large column weights. Two integer sequences entirely define the exponent matrix of the new method. The first sequence is an arithmetic sequence starting from 0 with a tolerance of 1, and the second sequence is a special sequence composed of integers that meet the constraint of the greatest common divisor (GCD). The existing explicit method can only provide a variety of row weight types with large circulant size. Compared with the recently proposed symmetric structure method based on computer search, the new explicit construction method has better decoding performance, extremely low description complexity, and does not require computer search.
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.
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.