Design of good QC-LDPC codes without small girth in the p-plane

Lingjun Kong, Yang Xiao

A construction method based on the p-plane to design high-girth quasi-cyclic low-density parity-check (QC-LDPC) codes is proposed. Firstly the good points in every line of the p-plane can be ascertained through filtering the bad points, because the designed parity-check matrixes using these points have the short cycles in Tanner graph of codes. Then one of the best points from the residual good points of every line in the p-plane will be found, respectively. The optimal point is also singled out according to the bit error rate (BER) performance of the QC-LDPC codes at last. Explicit necessary and sufficient conditions for the QC-LDPC codes to have no short cycles are presented which are in favor of removing the bad points in the p-plane. Since preventing the short cycles also prevents the small stopping sets, the proposed construction method also leads to QC-LDPC codes with a higher stopping distance.

Hyunki Kim, Eugene Ha, Jeongseok Ha

Lingjun Kong, Yang Xiao

The existing construction of Quasi-Cyclic low-density parity-check (short for QC-LDPC) codes has not considered the problems of small stopping sets and small girth and small minimum code weight together, while their existences will lead to the BER performance of QC-LDPC codes to be much poorer than that of randomly constructed LDPC codes even decoding failure. To solve the problem, some theorems of the QC-LDPC codes without small stopping sets and small girth were proposed first. Then good QC-LDPC codes without small stopping sets and small girth were designed based on the row vectors of generator matrix. The small minimum code weight also can be prevented due to the new algorithm based on the row vectors of generator matrix, which is the new way to choose the shifted factors to construct the good QC-LDPC codes for the given prime. The effectiveness and the practicability of the algorithm are demonstrated by the simulation results. It is significant for us to analyze and optimize the design of LDPC codes.

Jianrong Bao, Dan He, Bin Jiang +1 more

Lingjun Kong, Yang Xiao

It is well known that the performance of low-density parity check (LDPC) codes under iterative decoding is determined by certain combinatorial structures (such as stopping sets and girth) in the Tanner graph. The difficulty in enumerating all possible combinations of the columns may prevent an efficient search of good LDPC codes with small stopping sets and girth. To solve the problem, this paper presents a new algorithm of counting the number of small stopping sets and girth by analyzing the shapes of the cycles of Tanner graph in parity check matrix for designing good LDPC codes which is less complex than the existing algorithms. This method can be used effectively to evaluate the performance of LDPC codes according to their small stopping sets and girth distributions.

Jasvinder Singh, Manish Gupta, Jaskarn Singh Bhullar