Regular Quasi-cyclic Low-density Parity-check Codes Research Articles

Some New Constructions of Girth-8 QC-LDPC Codes for Future GNSS

In this letter, we propose a new construction of girth-8 Quasi-Cyclic Low-Density Parity-Check codes (QC-LDPC) with various lengths for the global navigation satellite systems (GNSS). This scheme combines two steps. The first is the construction of a family of regular girth-8 QC-LDPC codes of various lengths and rates with two designed sequences. The second is the performance improvement of those from the first construction of half-rate cases using a proposed weight matrix so that the result becomes type-II QC-LDPC codes. This results in some final codes with short lengths of 600, 1200, and 1800, especially for future GNSS. We performed a simulation and confirmed that the proposed QC-LDPC codes of lengths 600 and 1200 have an additional coding gain about 0.3 dB at frame error rate 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−5</sup> over the LDPC codes used in the Global Positioning System.

FER Performance Analysis and Optimization of Diagonal Structure Based QC-LDPC Codes with Girth 12 Using LU Decomposition

The manuscript presents the construction of a diagonally structured Parity check matrix using lower and upper (LU) decomposition for regular Quasi Cyclic-Low Density Parity Check Codes (QC-LDPC) with girth 12. This diagonally structured parity check matrix reduces the sparsity characteristics of QC-LDPC codes. The regular Quasi Cyclic-LDPC code has been constructed using circulant shifting and randomization method in which uniform distribution of non-zero elements, XOR operation, and pairing of row and column has been done. A Simplified Log Domain Sum-Product Algorithm (SLDSPA) is used for the decoding of the constructed code. To reduce the CPU run time of the constructed code, a column-wise loop optimization technique is applied. The frame error rate and CPU run time is calculated for different code lengths such as code1 (816, 408), code2 (1008, 504), code3 (1032, 516), code4 (1944, 972), code5 (672, 336) and code6 (504, 252). The construction of regular QC-LDPC codes on half code rate is obtained for the weight of each row is 6 and each column is given by 3. The results showing that the constructed code is better performing in terms of Low FER (10−1 to 10−7), Low SNR (0 to 3.5 dB) and Less CPU run time in milliseconds.

A construction of the high-rate regular quasi-cyclic LDPC codes

In this paper, a scheme to construct the high-rate regular quasi-cyclic low-density parity-check (QC-LDPC) codes is developed based on the finite geometry low-density parity-check (LDPC) codes. To achieve this, we first decompose the EG-LDPC code into a set of the block submatrices with each of them being cyclic. Utilizing the decomposed structure, we then construct an auxiliary matrix to facilitate us for identifying and eliminating 6-cycles of the block submatrices. In the simulation, seven regular QC-LDPC codes with code rates being 4/5, 7/8, 10/11, 11/12, and 12/13 of moderate code lengths are constructed with this scheme. The performances of these codes demonstrate the effectiveness of the proposed scheme.

Analytical Lower Bound on the Lifting Degree of Multiple-Edge QC-LDPC Codes With Girth 6

Multiple-edge protographs have some advantages over single-edge protographs, such as potentially having larger minimum Hamming distance. However, most of results in the literature are related to the construction of single-edge quasi-cyclic low-density parity-check codes (QC-LDPC) codes and little research has been done for the construction of multiple-edge QC-LDPC codes. In this letter, for the first time, necessary and sufficient conditions for the exponent matrices to have multiple-edge QC-LDPC codes with girth 6 are provided. As a consequence of this letter, a lower bound on the lifting degree of regular and irregular multiple-edge QC-LDPC codes with girth 6 is derived. We also present QC-LDPC codes whose lifting degrees meet our proposed lower bound. These codes have shorter lengths compared with single-edge QC-LDPC codes. Another contribution of this letter is presenting a technique to reduce the size of the search space to find these codes.

Explicit Constructions for Type-1 QC-LDPC Codes With Girth 12

Given any $J\times J\,\,(J>3)$ square matrix over $\textbf {Z}_{P}$ such that the differences of any two row vectors contain each element in $\textbf {Z}_{P}$ at most once, a class of $(3,L)$ -regular quasi-cyclic low-density parity-check codes is explicitly constructed with lengths $PJL^{2}$ and girth 12, where $L$ is any integer satisfying $3 . Simulation results show that the new codes perform very well for moderate rates and lengths.

A novel construction method of QC-LDPC codes based on CRT for optical communications

A novel construction method of quasi-cyclic low-density parity-check (QC-LDPC) codes is proposed based on Chinese remainder theory (CRT). The method can not only increase the code length without reducing the girth, but also greatly enhance the code rate, so it is easy to construct a high-rate code. The simulation results show that at the bit error rate (BER) of 10−7, the net coding gain (NCG) of the regular QC-LDPC(4 851, 4 546) code is respectively 2.06 dB, 1.36 dB, 0.53 dB and 0.31 dB more than those of the classic RS(255, 239) code in ITU-T G.975, the LDPC(32 640, 30 592) code in ITU-T G.975.1, the QC-LDPC(3 664, 3 436) code constructed by the improved combining construction method based on CRT and the irregular QC-LDPC(3 843, 3 603) code constructed by the construction method based on the Galois field (GF(q)) multiplicative group. Furthermore, all these five codes have the same code rate of 0.937. Therefore, the regular QC-LDPC(4 851, 4 546) code constructed by the proposed construction method has excellent error-correction performance, and can be more suitable for optical transmission systems.

Symmetrical Constructions for Regular Girth-8 QC-LDPC Codes

In this paper, we propose new constructions for regular girth-8 quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices (CPM). The constructions assume symmetries in the structure of the parity-check matrix and employ a greedy exhaustive search algorithm to find the permutation shifts of the CPMs. As a result of symmetries, the new codes have a more compact representation compared with their counterparts. In majority of cases, also, they achieve the girth 8 at a shorter block length for the same degree distribution (code rate). Deterministic (explicit) constructions are also presented to expand the proposed parity-check matrices to larger block lengths and higher rates. The proposed long high-rate codes are often substantially shorter than regular girth-8 QC-LDPC codes of similar rate in the literature. Simulation results demonstrate that the proposed symmetric codes have competitive performance in comparison with similar existing QC-LDPC codes that lack symmetry.

Design and Analysis of Root-Protograph LDPC Codes for Non-Ergodic Block-Fading Channels

We investigate the performance of the protograph low-density parity-check (LDPC) codes over Nakagami block-fading (BF) channels with multiple receive antennas. Using the modified protograph extrinsic information transfer (PEXIT) algorithm, we observe that the existing protograph LDPC codes, which have been shown to possess excellent error performance over additive white Gaussian noise channels, cannot perform well over non-ergodic BF channels. To address this problem, a simple design scheme is proposed to construct three new root-protograph (RP) LDPC codes, i.e., the regular RP code, the improved RP1 (IRP1) code, and the IRP2 code. The convergence analysis, error-performance analysis, and simulated results show that the three proposed RP codes outperform the existing protograph LDPC codes and regular quasi-cyclic LDPC code. Furthermore, the IRP1 code and the IRP2 code are superior to the regular RP code and the regular root-LDPC code and exhibit outage-limit-approaching error performance. Consequently, these two proposed IRP codes appear to be better alternatives as compared to other error-correction codes for slowly-varying wireless communication systems.

A new coding scheme of QC-LDPC codes for optical transmission systems

Based on Galois Field (GF(q)) multiplicative group, a new coding scheme for Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes is proposed, and the new coding scheme has some advantages such as the simpler construction, the easier implementation encoding, the lower complexity of the encoding and decoding, the more flexible adjustment of the code length as well as the code rate and so forth. Under the condition of considering the characteristics of optical transmission systems, an irregular QC-LDPC (3843,3603) code to be suitable for optical transmission systems is constructed by applying the proposed new coding scheme. The simulation result shows that the net coding gain (NCG) of the irregular QC-LDPC (3843,3603) code is respectively improved 2.14dB, 1.19dB, 0.24dB and 0.14dB more than those of the classic RS (255,239) code in ITU-T G.975, the LDPC (32640,30592) code in ITU-T G.975.1, the regular SCG-LDPC (3969,3720) code constructed by the Systematically Constructed Gallager (SCG) coding scheme and the regular QC-LDPC (4221,3956) code at the bit error rate (BER) of 10-8. Furthermore, all the five codes have the same code rate of 93.7%. Therefore, the irregular QC-LDPC (3843,3603) code constructed by the proposed new coding scheme has the more excellent error-correction performance and can be better suitable for optical transmission systems.

A novel QC-LDPC code based on the finite field multiplicative group for optical communications

A novel construction method of quasi-cyclic low-density parity-check (QC-LDPC) code is proposed based on the finite field multiplicative group, which has easier construction, more flexible code-length code-rate adjustment and lower encoding/decoding complexity. Moreover, a regular QC-LDPC(5334,4962) code is constructed. The simulation results show that the constructed QC-LDPC(5334,4962) code can gain better error correction performance under the condition of the additive white Gaussian noise (AWGN) channel with iterative decoding sum-product algorithm (SPA). At the bit error rate (BER) of 10−6, the net coding gain (NCG) of the constructed QC-LDPC(5334,4962) code is 1.8 dB, 0.9 dB and 0.2 dB more than that of the classic RS(255,239) code in ITU-T G.975, the LDPC(32640,30592) code in ITU-T G.975.1 and the SCG-LDPC(3969,3720) code constructed by the random method, respectively. So it is more suitable for optical communication systems.

4-Cycle Free LDPC Codes Based on Difference Sets

Difference sets are used to give a novel method for constructing structured regular quasi-cyclic low-density parity-check codes. Let Fq be a field with q elements and assume that D={d1,..., dk} is a (v,k,1)-difference set for Zv with d1<;d2<;...<;dk. Depending on v=q-1, v≥ 2dk or v≥ 2dk-1, three code construction methods are given that produce regular 4-cycle free codes, that is codes with girth at least 6. Simulation results show that the constructed codes perform well over the AWGN channel with an iterative sum-product message-passing decoding algorithm.

Construction of High-Rate Regular Quasi-Cyclic LDPC Codes Based on Cyclic Difference Families

For a high-rate case, it is difficult to randomly construct good low-density parity-check (LDPC) codes of short and moderate lengths because their Tanner graphs are prone to making short cycles. Also, the existing high-rate quasi-cyclic (QC) LDPC codes can be constructed only for very restricted code parameters. In this paper, a new construction method of high-rate regular QC LDPC codes with parity-check matrices consisting of a single row of circulants with the column-weight 3 or 4 is proposed based on special classes of cyclic difference families. The proposed QC LDPC codes can be constructed for various code rates and lengths including the minimum achievable length for a given design rate, which cannot be achieved by the existing high-rate QC LDPC codes. It is observed that the parity-check matrices of the proposed QC LDPC codes have full rank. It is shown that the error correcting performance of the proposed QC LDPC codes of short and moderate lengths is almost the same as that of the existing ones through numerical analysis.

Coset graphs for low-density parity check codes: performance on the binary erasure channel

The authors show that a popular way of constructing quasi-cyclic low-density parity check (LDPC) codes is a special case of a construction which is common in graph theory and group theory. It is shown that a generalisation of this construction as coset graphs produces (dv, dc)-regular LDPC codes that have an advantage in terms of the minimum stopping set size compared to quasi-cyclic LDPC codes. A (dv, dc)-regular quasi-cyclic LDPC code cannot have minimum stopping set size larger than (dv+1)!. However, by using coset graphs, a (3, 5)-regular LDPC code with minimum stopping set size of 28 and a (3, 4)-regular LDPC code with minimum stopping set size larger than 32 have been obtained. In addition, the idea of coset graphs also provides a compact algebraic way of describing bipartite graph and the associated parity-check matrix of an LDPC code. Simulation results of iterative decoding of the coset graphs LDPC codes over the binary erasure channel show that some of the codes converge well and based on the truncated stopping set distributions of the codes, which are exhaustively and efficiently enumerated, the error-floor of the codes at low probability of erasure is estimated.

An Area-Efficient and Low-Power Multirate Decoder for Quasi-Cyclic Low-Density Parity-Check Codes

The quasi-cyclic low-density parity-check (QC-LDPC) codes are widely applied in digital broadcast and communication systems. However, the decoders are still difficult to be put into practice due to their large area and high power, especially in the wireless mobile devices. This paper presents an improved all-purpose multirate iterative decoder architecture for QC-LDPC codes, which can largely reduce their area and power. The architecture implements the normalized min-sum algorithm, rearranges the original two-phase message-passing flow, and adopts an efficient quantization method for the second minimum absolute values, an optimized storing scheme for the position indexes and signs, and an elaborate clock gating technique for substantive memories and registers. It is also configurable for any regular and irregular QC-LDPC codes, and can be easily tuned up to different code rates and code word lengths. The chip is fabricated in an SMIC 0.18- six-metal-layer standard CMOS technology. It attains a throughput of 104.5 Mb/s, and dissipates an average power of 486 mW at 125 MHz, and 15 decoding iterations. The core area is only 9.76 mm2. The chip has been applied into the China digital terrestrial/television multimedia broadcasting system.

High-Rate Nonbinary Regular Quasi-Cyclic LDPC Codes for Optical Communications

The parity-check matrix of a nonbinary (NB) low-density parity-check (LDPC) code over Galois field GF(q) is constructed by assigning nonzero elements from GF(q) to the 1s in corresponding binary LDPC code. In this paper, we state and prove a theorem that establishes a necessary and sufficient condition that an NB matrix over GF(q), constructed by assigning nonzero elements from GF(q) to the 1s in the parity-check matrix of a binary quasi-cyclic (QC) LDPC code, must satisfy in order for its null-space to define a nonbinary QC-LDPC (NB-QC-LDPC) code. We also provide a general scheme for constructing NB-QC-LDPC codes along with some other code construction schemes targeting different goals, e.g., a scheme that can be used to construct codes for which the fast-Fourier-transform-based decoding algorithm does not contain any intermediary permutation blocks between bit node processing and check node processing steps. Via Monte Carlo simulations, we demonstrate that NB-QC-LDPC codes can achieve a net effective coding gain of 10.8 dB at an output bit error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-12</sup> . Due to their structural properties that can be exploited during encoding/decoding and impressive error rate performance, NB-QC-LDPC codes are strong candidates for application in optical communications.