Construction of LDPC codes

Huang Shen

This paper presents a construction method of quasi-cyclic non-binary low density parity-check(NB-LDPC)code based on the finite field three-dimensional construction.After construction of base matrix based on the finite field,a cyclic matrix is constructed with the method of three-dimensional extension.Finally,non-binary parity check matrix with quasi-cyclic feature is constructed.Compared with the GF(29)-based RS(511,305)and NB-LDPC code constructed by the three-dimensional construction can increase 3.3dB net code gain(NCG)at the bit error rate(BER)of 10-7.Compared with the binary QC-LDPC codes constructed by the traditional way,the binary LDPC codes randomly constructed,the32-ary QC-LDPC codes constructed based on finite fields and the 64-ary cyclic LDPC codes constructed based on Euclidean geometry,the NB-LDPC code constructed by the three-dimensional construction can increase 0.56dB,0.56dB,0.03dB and 0.83dB NCGrespectively at the BERof 10-7.Through performance simulation of the NB-LDPC code constructed based on this three-dimensional construction,it shows that such an NB-LDPC code not only has quasi-cyclic feature,conducive to hardware implementation,but also shows better error correction performance in the short code length.

Vikram Arkalgud Chandrasetty, Syed Mahfuzul Aziz

Farzaneh Abedi, Mohammad Gholami

Multi-type quasi-cyclic (QC) low-density parity-check (LDPC) codes can be considered as multiple-edge protograph QC-LDPC codes having some advantages in the minimum Hamming distance bound over single-edge protograph codes or type-I QC-LDPC codes when the base matrices have the same size. In this paper, we investigate a class of multi-type QC-LDPC codes whose parity-check matrices contain just one blockrow of circulants and we obtain the generator matrix of such codes in general form. Using the permutation arrays and defining injection arrays, we present a new approach to construct a class of high-rate type-I QC-LDPC codes with girth 6 from the constructed 4-cycle free multi-type QC-LDPC codes. In continue, for 2 ≤ w ≤ 6, some type-w QC-LDPC codes with girth 6 are constructed explicitly such that the constructed codes are flexible in terms of rate and length. To the best of our knowledge, for w = 5,6, this is the first paper which deals with the explicit construction of type-w QC-LDPC codes with girth 6 and high rates. Moreover, for w = 3,4, the constructed type-w QC-LDPC codes have better (6,8)-cycle multiplicities than the codes with minimum achievable length recently constructed by cyclic difference families (CDFs). Simulation results show that the binary and non-binary constructed codes outperform the constituent underlying QC-LDPC codes.

W.E. Ryan

Mohammad Gholami, Masoumeh Alinia

Low‐density parity‐check (LDPC) codes from affine permutation matrices , called APM‐LDPC codes, are a class of LDPC codes whose parity‐check matrices consist of zero matrices or APMs of the same orders. APM‐LDPC codes are not quasi‐cyclic (QC), in general. In this study, necessary and sufficient conditions are provided for an APM‐LDPC code to have cycles of length 2 l , l ≥ 2, and a deterministic algorithm is given to generate APM‐LDPC codes with a given girth. Unlike Type‐I conventional QC‐LDPC codes, the constructed ( J , L ) APM‐LDPC codes with the J × L all‐one base matrix can achieve minimum distance greater than ( J + 1)! and girth larger than 12. Moreover, the lengths of the constructed APM‐LDPC codes, in some cases, are smaller than the best known lengths reported for QC‐LDPC codes with the same base matrices. Another significant advantage of the constructed APM‐LDPC codes is that they have remarkably fewer cycle multiplicities compared with QC‐LDPC codes with the same base matrices and the same lengths. Simulation results show that the binary and non‐binary constructed APM‐LDPC codes with lower girth outperform QC‐LDPC codes with larger girth.