On Graph Constructions for LDPC Codes by Quasi-Cyclic Extension

Qiuju Diao, Qin Huang, Shu Hwa Lin +1 more

An approach for studying quasi-cyclic codes based on matrix transformations via Fourier transforms and row and column permutations is presented. These transformations put a parity-check matrix in the form of an array of circulant matrices into a diagonal array of matrices of the same size over an extension field. The approach is used to characterize certain structural properties of low-density parity-check (LDPC) codes such as the girths of their Tanner graphs. Many constructions of quasi-cyclic LDPC codes can be unified under the proposed approach.

Xiao Xiao, Lee

H. Pishro-Nik, Faramarz Fekri

Jialing Li, Wen Chen, Xueqin Jiang +2 more

This paper proposes a new approach to construct quasi-cyclic low-density parity-check (QC-LDPC) codes of girth 8. The proposed codes are based on a bipartite graph called Moore graph. By decomposing the bit nodes of the bipartite graph, the proposed QC-LDPC codes have variable code lengths and rates. Simulation results show that the proposed QC-LDPC codes have good performance with an iterative decoding over an AWGN channel.

Chao Chen

Josef Lauri, Cen Jung Tjhai

The authors show that a popular way of constructing quasi-cyclic low-density parity check (LDPC) codes is a special case of a construction which is common in graph theory and group theory. It is shown that a generalisation of this construction as coset graphs produces (dv, dc)-regular LDPC codes that have an advantage in terms of the minimum stopping set size compared to quasi-cyclic LDPC codes. A (dv, dc)-regular quasi-cyclic LDPC code cannot have minimum stopping set size larger than (dv+1)!. However, by using coset graphs, a (3, 5)-regular LDPC code with minimum stopping set size of 28 and a (3, 4)-regular LDPC code with minimum stopping set size larger than 32 have been obtained. In addition, the idea of coset graphs also provides a compact algebraic way of describing bipartite graph and the associated parity-check matrix of an LDPC code. Simulation results of iterative decoding of the coset graphs LDPC codes over the binary erasure channel show that some of the codes converge well and based on the truncated stopping set distributions of the codes, which are exhaustively and efficiently enumerated, the error-floor of the codes at low probability of erasure is estimated.