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

Quasi-Cyclic Spatially Coupled LDPC Codes Construction with Girth Eight

Shikui Guo, Guohua Zhang

Published Aug 10, 2025• 0 citations
Publisher

Abstract

Small cycles in spatially coupled low-density paritycheck (SC-LDPC) codes have a detrimental effect on the error floor. Recently, an explicit simple method has been proposed to construct a class of 4-cycle free SC-LDPC codes by partitioning a spreading matrix into a number of component matrices. Using a quasi-cyclic (QC) lifting, we propose a component matrices construction method to obtain QC-SC-LDPC codes with a girth eight in this paper. The component matrices of the new method are completely defined by a spreading matrix and an exponent matrix. The spreading matrix is constructed from a finite sequence of integers, and the exponent matrix is constructed from the greatest common divisor (GCD) framework. The novel construction method demonstrates the capacity to entirely eliminate 6-cycles. Compared to the recently proposed explicit structural design, the proposed code achieves a girth of eight and exhibits better bit-error rate (BER) performance.

DOI
10.1109/icccworkshops67136.2025.11148217

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Small cycles in spatially coupled low-density paritycheck (SC-LDPC) codes have a detrimental effect on the error floor. Recently, an explicit simple method has been proposed to construct a class of 4-cycle free SC-LDPC codes by partitioning a spreading matrix into a number of component matrices. Using a quasi-cyclic (QC) lifting, we propose a component matrices construction method to obtain QC-SC-LDPC codes with a girth eight in this paper. The component matrices of the new method are completely defined by a spreading matrix and an exponent matrix. The spreading matrix is constructed from a finite sequence of integers, and the exponent matrix is constructed from the greatest common divisor (GCD) framework. The novel construction method demonstrates the capacity to entirely eliminate 6-cycles. Compared to the recently proposed explicit structural design, the proposed code achieves a girth of eight and exhibits better bit-error rate (BER) performance.
Spatially coupled (SC) low-density parity-check (LDPC) codes can achieve capacity approaching performance with low message recovery latency when using sliding window (SW) decoding. An SC-LDPC code constructed from a protograph can be generated by first coupling a chain of block protographs and then lifting the coupled protograph using permutation matrices. In this paper, we introduce a systematic design to eliminate 4-cycles in a coupled protograph. Further using a quasi-cyclic (QC) lifting, we introduce a procedure for constructing QC-SC-LDPC codes of girth at least eight. This can be interpreted as a multi-stage graph lifting process that yields a greater flexibility in designing QC-SC-LDPC codes with a large girth than previous approaches. Simulation results show the design leads to improved decoding performance, particularly in the error floor, compared to random constructions. Finally, we determine the minimum coupling width required to eliminate 4-cycles in a coupled protograph.
Quasi-Cyclic (QC) low-density parity-check (LDPC) codes are a class of LDPC codes with a simple construction facilitating hardware implementation while achieving excellent performance. In this paper, we introduce an algorithm that constructs QC spatially-coupled (SC) LDPC codes with large girth while keeping the constraint length small. The algorithm offers a "protograph to basegraph" construction, focusing on finding small lifting sizes of QC codes while avoiding short cycles. This work extends the hierarchical quasi-cyclic (HQC) construction for block LDPC codes proposed by Wang et al. to the spatially coupled case. The construction is based on the cycle relevant matrix (CRM) derived from the periodic structure of time-invariant SC-LDPC codes. Numerical results show that the proposed algorithm effectively achieves the target girth with a small lifting factor, enabling low-complexity SC code construction.
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.
Recently, a class of low-density parity-check codes based on affine permutation matrices, called APM-LDPC codes, have been considered which have some advantages than quasi-cyclic (QC) LDPC codes in terms of minimum-distance, cycle distribution, and error-rate performance. Moreover, some explicit constructions for exponent matrices of conventional APM-LDPC codes with girth at least 6 have been investigated. In this paper, a class of 4-cycle free APM-LDPC codes is constructed by a new explicit method such that the constructed codes have better cycle distributions rather than the recently proposed APM codes with girth 6. As simulation results show, the constructed codes outperform PEG and random-like LDPC codes with the same rates and lengths.
In this paper, we deal with time-varing quasi-cyclic (QC) spatially-coupled (SC) low-density parity-check (LDPC) codes by lifting the coupled protographs. The construction method of girth-6 protographs for any code rate is proposed. We provide a theoretical lower bound on the syndrome former constraint length under condition of the protographs without 4-cycles in their Tanner graphs. Then we use periodic QC lifting method to search the QC-SC-LDPC matrices with larger girth based on the protographs. The minimum dimension size of QC-SC-LDPC codes is given to count the girth of QC-SC-LDPC codes with a semi- infinite binary-check matrix structure. The performance of such constructed codes with lower complexity and delay is superior to the LDPC block codes.
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.