A mathematical tool for constructing parametrizable spatially-coupled LDPC codes with cyclic structure and large girth

宇 梁

Haizheng Li, Sisi Miao, Laurent Schmalen

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.

Markus Stinner

Shiyuan Mo, Li Chen, Daniel J. Costello +3 more

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.

Valentina Pepe

Shuoshuo Wang, Zhanji Wu, Qihao Wu

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.