LDPC codes with large girth based on voltage graphs

Kenya Sugihara, Hideo Yoshida

Mahnaz Ahmadi, Hadi Dehghani, ‎Saeid Alikhani +1 more

Srinivas Subramanian, Asit Kumar Pradhan, Andrew Thangaraj

Low Density Parity Check (LDPC) codes have capacity-approaching performance over several channels of interest. In practice, for good block-error rate performance, the girth of the Tanner graph of an LDPC code needs to be as high as possible. In theory, to show that block-error rate approaches zero for increasing block-lengths, the girth of the Tanner graph sequence needs to tend to infinity with block-length. To meet these requirements, we construct sequences of large-girth irregular LDPC codes for a given degree-distribution pair (DDP) by applying a general node splitting algorithm on large girth graphs. The obtained Tanner graph meets the required DDP up to a suitable approximation. By optimizing the node-splitting method and using suitable large-girth graphs, we show code constructions with smaller block length for the same girth, when compared to previous constructions. Similar gains in block length are observed in the construction of sequences of large-girth protograph LDPC codes. Simulations, over a binary erasure channel, confirm the gains in block-error rate obtained by the large girth construction.

H. Pishro-Nik, Faramarz Fekri

Hong Zhou

The girth is an indication of the quality of Low Density Parity Check(LDPC) Codes. For any given LDPC Codes,quickly checking their girths is of some important significance. An algorithm for quickly checking the girth of LDPC Codes in bipartite graphs is presented. This algorithm can also give out the numbers of the shortest cycles passing through the given check node (or bit node). The testing result of this algorithm implemented in MATLAB shows that,even for large-scale H matrix of LDPC Codes like PEGReg504x1008,it takes only 2.876 s for the algorithm to finish checking.

Xueqin Jiang, Han Hai, Hui-Ming Wang +1 more

For a given base graph, the lifted graph can be obtained by a copy-and-permute procedure. If the permutation is cyclic, the lifted graph corresponds to a quasi-cyclic (QC) protograph low-density parity-check (LDPC) code. The girth of the QC protograph LDPC code is determined by the girth of the base graph and the permutation shifts. In this paper, we first derive a lower bound on the lifting degree to achieve a large girth lifted graph. Then, motivated by the cycle searching and girth maximizing features of the progressive edge-growth (PEG) algorithm, we introduce the permutation shifts determining (PSD) PEG algorithm, which can construct large girth base graph and determine the optimal permutation shifts, simultaneously. It is shown that the computational complexity of PSD-PEG algorithm is much lower than that of the PEG algorithm and the PEG-QC algorithm for the same codeword length. Furthermore, we show that the PSD-PEG algorithm can also be used to construct nonbinary QC protograph LDPC codes without low weight codes. Simulation results show that the binary and nonbinary QC protograph LDPC codes constructed by the PSD-PEG algorithm have good bit error rate performance and frame error rate performance over the additive white Gaussian noise channel.