In this paper we present a class of structured low-density parity-check (LDPC) codes from affine permutation matrices, called the APM-LDPC codes, which are a generalization of quasi-cyclic LDPC codes. We give a necessary and sufficient condition under which an APM-LDPC code has a cycle and introduce a simple method to construct APM-LDPC codes of large length by combining those of small length based on the Chinese remainder theorem. In particular, we show that the girth of APM-LDPC codes obtained in this method is always larger than or equal to those of given APM-LDPC codes
In this paper we present a class of structured low-density parity-check (LDPC) codes from affine permutation matrices, called the APM-LDPC codes, which are a generalization of quasi-cyclic LDPC codes. We give a necessary and sufficient condition under which an APM-LDPC code has a cycle and introduce a simple method to construct APM-LDPC codes of large length by combining those of small length based on the Chinese remainder theorem. In particular, we show that the girth of APM-LDPC codes obtained in this method is always larger than or equal to those of given APM-LDPC codes
Recently, Low Density Parity-Check (LDPC) codes based on Affine Permutation Matrices (APM) drew lots of attention. Compared with the Quasi-Cyclic LDPC (QC-LDPC) codes, these kinds of codes have some advantages. APM-LDPC codes obtain lower cycle-distributions, minimum hamming distance and greater girth. This paper explains the importance of cyclic distribution by comparing APM-LDPC codes with QC-LDPC codes. Then a particular form of APM-LDPC codes is proposed and researched. The new codes can low down the cycle-distribution to larger extent. In the following research, an effective method, which constructs the new codes with fixed girth, is proposed. Simulations show that the construction method is reasonable and effective. The transmission performances are better than the traditional methods, as well. Finally, the implementation and verification are carried out on FPGA.
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.
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.
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 we propose a method of constructing quasi-cyclic low-density parity-check (QC-LDPC) codes of large length by combining QC-LDPC codes of small length as their component codes, via the Chinese remainder theorem. The girth of the QC-LDPC codes obtained by the proposed method is always larger than or equal to that of each component code. By applying the method to array codes, we present a family of high-rate regular QC-LDPC codes with no 4-cycles. Simulation results show that they have almost the same performance as random regular LDPC codes.
Low-density parity-check (LDPC) codes based on affine permutation matrices, APM-LDPC codes, have been attracted recently, because of some advantages rather than QC-LDPC codes in minimum-distance, cycle distribution and error-rate performance. In this letter, circulant and anti-circulant permutation matrices are used to define a class of LDPC codes, called AQC-LDPC codes, which can be considered as an special case of APM-LDPC codes. In fact, each AQC-LDPC code can be verified by a sign matrix and a slope matrix which are helpful to show each cycle in the Tanner graph by a modular linear equation. For the normal sign matrix A , if -1 ϵ, A, it is shown that the corresponding AQC-LDPC code has maximum-girth 8. Finally, two explicit constructions for AQC-LDPC codes with girths 6, 8 are presented which have some benefits rather than the explicitly constructed QC and APM LDPC codes in minimum-distance, cycle distributions and bit-error-rate performances.