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.
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.
Multitype quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of protograph LDPC codes lifted cyclically from protographs with multiple edges, represented by two weight and slope matrices. For a given weight-matrix, an approach is proposed to find the maximum-achievable girthgmaxof the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girthgmax, some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girthg,g≤gmax, have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.
In this paper, we consider the constructions of type-II quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight from a Sidon sequence. We first derive the necessary and sufficient conditions guaranteeing a girth-eight type-II QC-LDPC code. By combining these conditions with the concept of the Sidon sequence, three classes of type-II QC-LDPC codes are subsequently proposed with girth eight. To the best of our knowledge, the second and the third classes we proposed are the first series of systematic and algebraic constructions for the girth-eight type-II QC-LDPC codes that have rates being greater than a half and, at the same time, have distance upper bounds being not limited to 12. Via simulations, we show the promising performance of the proposed type-II QC-LDPC codes with girth eight. In addition, a set of general bounds and explicit/random constructions without using a Sidon sequence are also presented for the type-II QC-LDPC codes with girth six or eight. We observe that the proposed codes perform better than or equally well as the random QC-LDPC codes from PEG and quadr.-congr. methods, while the novel codes possess both highly structured parity-check matrices and very flexible choices in circulant size.
A simple and general definition of quasi cyclic low density parity check (QC LDPC) codes which are constructed based on circulant permutation matrices (CPM) is proposed. As an special case of this definition, we first represent one type of so called combinatorially designed multiple edge protograph codes. The code construction is mainly based on perfect difference families (PDF) and is called Construction 1. Secondly, using the proposed Construction 1 along with a technique named as column dispersion technique (CDT), we design several types of multiple-edge CPM QC LDPC codes (codes with Construction 2) in a wide range of rates, lengths, girths and minimum distances. Parameters of some of these codes are reported in tables. Also included in this paper are the multiplicities of short simple cycles of length up to 10 in Tanner graph of our constructed codes. Our experimental results for short to moderate length codes show that although minimum distances of codes play an important role in waterfall region, the higher the number of short simple cycles is, the better (sharper) the waterfall is. The performances of many codes such as WiMAX, PEG, array, MacKay, algebraic and combinatorial, and also, symmetrical codes have compared with our constructed codes. Based on our numerical results and regardless of how a code is constructed, those with higher number of short simple cycles and higher minimum distances, have better waterfalls.
Recently, some attentions have been focused on a class of low-density parity-check codes from affine permutation matrices, called APM-LDPC codes, which have some advantages rather than quasi-cyclic low-density parity-check (QC-LDPC) codes in the minimum-distance, cycle distribution and error-rate performance. In this letter, first, some novel explicit constructions for exponent matrices of conventional APM-LDPC codes with girths 6, 8, and 10 are investigated, and for each given exponent matrix E corresponding to an APM-LDPC code with girth 2g , 3 ≤ g ≤ 5, a lower bound Q = Q(E,g) is obtained, such that APM-LDPC codes with exponent matrix E have girth at least 2g, for each (prime) APM size m ≥ Q. Then, regarding to reduce the lower bound, an improved bound IQ is obtained, which is significantly better than some lower bounds reported for QC-LDPC codes with explicit constructions. Simulation results show that the constructed APM-LDPC codes outperform QC-LDPC codes with the same girth.
In this paper a new low complexity method for constructing binary quasi-cyclic low-density parity-check (QC-LDPC) codes is introduced. In the proposed method, each block-column of the parity check matrix H is made by a circulant matrix in a way that the associated Tanner graph is free of cycle four. Each circulant matrix in H is made by a generator column. The generator columns should be selected in a way that each associated circulant matrix and every two distinct circulant matrices are free of cycle four. The generator columns are made by row distance sets. An algorithm for generating distance sets and obtaining circulant matrices with columns of weight three is presented separately. Simplicity of construction and having a good flexible family of quasi cyclic LDPC codes both in rate and length are the main properties of the proposed method. The performance of the proposed codes is compared with that of the random-like and Array LDPC codes over an AWGN channel. Simulation results show that from the performance perspective, the constructed codes are competitive with random-like and Array LDPC codes.
Recently, a class of low-density parity-check codes based on affine permutation matrices, called APM-LDPC codes, have been considered which have some advantages than quasi-cyclic (QC) LDPC codes in terms of minimum-distance, cycle distribution, and error-rate performance. Moreover, some explicit constructions for exponent matrices of conventional APM-LDPC codes with girth at least 6 have been investigated. In this paper, a class of 4-cycle free APM-LDPC codes is constructed by a new explicit method such that the constructed codes have better cycle distributions rather than the recently proposed APM codes with girth 6. As simulation results show, the constructed codes outperform PEG and random-like LDPC codes with the same rates and lengths.