Construction Of Low-density Parity-check Codes Research Articles

On Construction of Low-Density Parity-Check Codes for Ultra-Reliable and Low Latency Communications

On Construction of Low-Density Parity-Check Codes for Ultra-Reliable and Low Latency Communications

Free-Ride Coding for Constructions of Coupled LDPC Codes

Free-ride coding, as an approach that admits transmission of a few extra bits over a low-density parity-check (LDPC) coded link without bandwidth expansion, is applied in this paper to construct coupled LDPC codes. Firstly, we present a syndrome channel model and derive the lower and upper bounds on its capacity (referred to as accessible capacity), indicating the feasibility of the reliable transmission of extra bits. Secondly, we present the performance evaluation on both the word error rate (WER) and the bit error rate (BER) for the free-ride codes with simple lower and upper bounds. Then we propose three applications of free-ride coding to construct coupled LDPC codes, including implicit globally-coupled LDPC (GC-LDPC) codes, partial product-LDPC codes, and terminated spatially-coupled LDPC (SC-LDPC) codes, all of which have the figure of merits that they share the same code rates with the basic component LDPC codes. Simulation results show that: 1) the proposed GC-LDPC codes can outperform the component LDPC codes, yielding a coding gain of up to 0.8 dB; 2) the proposed product codes with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(3,6)$ </tex-math></inline-formula> -regular LDPC component codes of length 1024 can lower the WER from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-2}$ </tex-math></inline-formula> down to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-6}$ </tex-math></inline-formula> at the SNR around 2 dB; 3) the proposed terminated SC-LDPC codes can perform as well as the conventional terminated SC-LDPC codes but without any rate loss.

Construction of Multi-Rate Quasi-Cyclic LDPC Codes for Satellite Communications

To provide reliable transmissions with flexible rates for satellite communications, this paper presents a novel method of constructing multi-rate quasi-cyclic low-density parity-check (LDPC) codes. The basic idea is to generate low-rate codes from a high-rate mother code by combining shortening and extending, which ensures that the generated code family owns the same code length, in order to maintain the same frame structure. The code construction involves the design of base matrices and exponent matrices for the designed codes. A progressive row elimination and addition algorithm is proposed for designing the code base matrices from a high rate to low rates. This algorithm leads to the nested and systematic structure of the parity-check matrices, which are desirable for practical implementations of their encoders and decoders, while ensuring the optimal decoding thresholds. In addition, we construct a circulation coefficient matrix based on finite fields and select the optimal rows in this matrix to construct exponent matrices while considering of cycle structures. We demonstrate that the designed codes achieve better performance for all the code rates than the LDPC codes in DVB-S2X standards. In addition, the proposed codes do not exhibit error floors for their block error rates down to 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−5</sup> .

Investigation of random-structure regular LDPC codes construction based on progressive edge-growth and algorithms for removal of short cycles

This paper investigates the construction of random-structure LDPC (low-density parity-check) codes using Progressive Edge-Growth (PEG) algorithm and two proposed algorithms for removing short cycles (CB1 and CB2 algorithm; CB stands for Cycle Break). Progressive Edge-Growth is an algorithm for computer-based design of random-structure LDPC codes, the role of which is to generate a Tanner graph (a bipartite graph, which represents a parity-check matrix of an error-correcting channel code) with as few short cycles as possible. Short cycles, especially the shortest ones with a length of 4 edges, in Tanner graphs of LDPC codes can degrade the performance of their decoding algorithm, because after certain number of decoding iterations, the information sent through its edges is no longer independent. The main contribution of this paper is the unique approach to the process of removing short cycles in the form of CB2 algorithm, which erases edges from the code's parity-check matrix without decreasing the minimum Hamming distance of the code. The two cycle-removing algorithms can be used to improve the error-correcting performance of PEG-generated (or any other) LDPC codes and achieved results are provided. All these algorithms were used to create a PEG LDPC code which rivals the best-known PEG-generated LDPC code with similar parameters provided by one of the founders of LDPC codes. The methods for generating the mentioned error-correcting codes are described along with simulations which compare the error-correcting performance of the original codes generated by the PEG algorithm, the PEG codes processed by either CB1 or CB2 algorithm and also external PEG code published by one of the founders of LDPC codes

A Rate Scalable Construction of Non-Homogeneous Quantum LDPC Codes of CSS Type Based on Balanced Incomplete Block Design

Balanced incomplete block design (BIBD) offers an important way to construct non-homogeneous quantum low-density-parity-check (LDPC) codes of Calderbank-Shor-Steane (CSS) type. However, the rates of the quantum codes based on this method are limited to be 1-2/m approximately, where m, the number of base blocks chosen from the associated BIBD, is even. We overcome this limitation by providing a rate scalable way to construct non-homogeneous quantum LDPC codes of CSS type. With our method, the rates of the resultant quantum codes are 1-2/m', where m'=m/e and e ∈ {1, 2}. Note that here m' can be odd. Thus, we double the rate range of the codes yielded from the existing strategies. Moreover, one can switch codes between the two rates flexibly adapting to various application scenarios.

Construction of QC LDPC Codes With Low Error Floor by Efficient Systematic Search and Elimination of Trapping Sets

We propose a systematic design of protograph-based quasi-cyclic (QC) low-density parity-check (LDPC) codes with low error floor. We first characterize the trapping sets of such codes and demonstrate, using edge coloring techniques, that the QC structure of the code eliminates some of the trapping set structures that can exist in a code with the same degree distribution and girth but lacking the QC structure. Based on this characterization, our design aims at eliminating a targeted collection of trapping sets. Considering the parent/child relationship between the trapping sets in the collection, we search for and eliminate those trapping sets that are in the collection but are not a child of any other trapping set in the collection. An efficient layered algorithm is designed for the search of these targeted trapping sets. Compared to the existing codes in the literature, the designed codes are superior in the sense that they are free of the same collection of trapping sets while having a smaller block length, or a larger collection of trapping sets while having the same block length. In addition, the efficiency of the search algorithm makes it possible to design codes with larger degrees which are free of trapping sets within larger ranges compared to the state-of-the-art.

New Construction of Low-Density Parity-Check Codes Based on Vector Space Over Finite Fields

Low-Density Parity-Check (LDPC) codes have low linear decoding complexity, which is a kind of good codes with excellent performance. Therefore, LDPC codes have great research value. Two kinds of LDPC codes are constructed based on vector space over finite field. The code length, code rate and minimum distance are given. Moreover, the two kinds of codes are compared with the existing codes, and the constructed codes are better than some existing ones in terms of code rate or minimum distance.

A Construction of High Performance Quasicyclic LDPC Codes: A Combinatoric Design Approach

This correspondence presents a construction of quasicyclic (QC) low-density parity-check (LDPC) codes based on a special type of combinatorial designs known as block disjoint difference families (BDDFs). The proposed construction of QC-LDPC codes gives parity-check matrices with column weight three and Tanner graphs having a girth lower-bounded by 6. The proposed QC-LDPC codes provide an excellent performance with iterative decoding over an additive white Gaussian-noise (AWGN) channel. Performance analysis shows that the proposed short and moderate length QC-LDPC codes perform as well as their competitors in the lower signal-to-noise ratio (SNR) region but outperform in the higher SNR region. Also, the codes constructed are quasicyclic in nature, so the encoding can be done with simple shift-register circuits with linear complexity.

Extended construction of array‐based non‐binary LDPC codes

Array-based non-binary (NB) low-density parity-check (LDPC) codes are attractive in practice for their advantages in implementation and performance. However, existing construction of array-based NB LDPC codes is only with the prime number cases. In this Letter, the authors extend the array-based construction to composite numbers to give an explicit construction of structured array-based NB LDPC codes with column weights three, four, and five. The resulting codes have girth at least six. The simulation results show that the proposed structured codes perform as well as the random codes constructed by progressive edge-growth algorithm, when decoded with the Q-ary sum product algorithm and the low-complexity weighted bit-reliability-based decoding algorithm.

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.

A Low-Complexity Hardware for Deterministic Compressive Sensing Reconstruction

Reconstruction algorithms of compressively sampled data include solving a sparse approximation problem. This problem requires iterative search or optimization techniques. Software implementations are not fast enough for real-time applications of recovery algorithms and these algorithms should be implemented in hardware. This paper presents a low-complexity hardware for real-time reconstruction of compressively sensed signal using orthogonal matching pursuit algorithm. The main goal of this research is to provide a deterministic alternative to the random measurement matrix. The construction of this matrix is based on the connection between the parity check matrix of low-density parity-check (LDPC) codes and the measurement matrix of compressive sensing. For efficient hardware realization, a geometric approach to the construction of LDPC codes is used for matrix generation on the fly without requiring a lot of storage. Cyclic and binary structure of this matrix leads to the lower computational complexity and hardware cost in the reconstruction side. The described hardware has been implemented and evaluated on a virtex6 field-programmable gate array from Xilinx. Implementation results show that the proposed architecture has better performance in terms of hardware utilization. Moreover, the accuracy of the recovered signal is comparable with the previous works in which the random measurement matrix is used.

On the Construction of Polar Codes for Channels With Moderate Input Alphabet Sizes

Current deterministic algorithms for the construction of polar codes can only be argued to be practical for channels with small input alphabet sizes. In this paper, we show that any construction algorithm for channels with moderate input alphabet size, which follows the paradigm of “degrading after each polarization step,” will inherently be impractical with respect to a certain “hard” underlying channel. This result also sheds light on why the construction of low-density parity-check codes using density evolution is impractical for channels with moderate-sized input alphabets.

Construction of Quasi-Cyclic LDPC Codes via Masking With Successive Cycle Elimination

In this letter, we study the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes by array dispersion and masking. We first propose a class of arrays of circulant permutation matrices (CPMs) of arbitrary size obtained by dispersion. Based on this class of arrays of CPMs, a class of QC-LDPC codes is constructed. For these resulting codes, we analyze their cycle structure and derive the numbers, which indicate how many cycles pass through each CPM in their parity-check matrices. Following the indication numbers with the order from largest to smallest, we replace the corresponding CPMs with zero matrices (ZMs) of the same size one by one until a targeted masking termination condition is satisfied. Then, the resultant arrays of CPMs and ZMs give a large class of QC-LDPC codes. Numerical results show that the codes constructed perform well over the additive white Gaussian noise (AWGN) channel when decoded with the sum-product algorithm.

Construction of High-Performance Array-Based Non-Binary LDPC Codes With Moderate Rates

Based on arrays, excellent low-density parity-check (LDPC) codes with high rates have been constructed. In this letter, we propose to construct high-performance nonbinary LDPC (NBLDPC) codes with moderate rates based on arrays. The proposed construction combines masking and column selection. First, we construct a masking matrix $\mathbf{M}$ based on asymptotic thresholds and certain error-prone substructures. Second, we generate a number of column selection sets (CSS) based on Golomb rulers. Based on $\mathbf{M}$ , we construct a parity-check matrix for each CSS. Third, we compute the cycle distribution of each parity-check matrix and select the one with the best cycle distribution as output. To assess the effectiveness of the proposed construction, error performances of several NBLDPC codes are presented. Comparisons with other codes are also carried out to show the advantages of our construction.

Superposition Construction of Q-ary LDPC Codes by Jointly Optimizing Girth and Number of Shortest Cycles

In this letter, we consider the construction of q-ary low-density parity-check (LDPC) codes by jointly optimizing the girth and number of shortest cycles in the frame of superposition. We begin with constructing binary quasi-cyclic LDPC codes by superposition. By replacing the nonzero entries of the parity-check matrices of the resulting binary codes with the nonzero elements of GF(q), a class of q-ary LDPC codes is obtained. Furthermore, we analyze the cycle structure of the constructed codes for a given degree distribution and code length, and present a search algorithm for finding q-ary LDPC codes with few cycles of length g, where g is the optimized girth value. Simulation results show that the proposed codes perform well under the iterative decoding algorithms.

Construction of Quantum LDPC Codes From Classical Row-Circulant QC-LDPCs

Classical row-circulant quasi-cyclic (QC) low-density parity check (LDPC) matrices are known to generate efficient high-rate short and moderate-length QC-LDPC codes, while the comparable random structures exhibit numerous short cycles of length-4. Therefore, we conceive a general formalism for constructing nondual-containing Calderbank–Shor–Steane (CSS)-type quantum low-density parity check (QLDPC) codes from arbitrary classical row-circulant QC-LDPC matrices. We apply our proposed formalism to the classical balanced incomplete block design (BIBD)-based row-circulant QC-LDPC codes for demonstrating that our designed codes outperform their dual-containing CSS-type counterparts as well as the entanglement-assisted (EA)-QLDPC codes.

Novel Construction and Optimization of LDPC Codes for NC-OFDM Cognitive Radio Systems

In this paper, a random Low-Density Parity-Check (LDPC) code is proposed for Noncontiguous Orthogonal Frequency Division Multiplexing (NC-OFDM) Cognitive Radio (CR) systems. Unlike other encoding schemes which achieve a system code rate of only 1/4 when half of the subcarriers are active, the proposed scheme achieves a system code rate of 1/2. The LDPC code is used to enhance the data transmission rate. A new channel model comprised of a binary erasure channel concatenated with an uncorrelated fading channel and an AWGN channel is adopted for NC-OFDM CR systems. Moreover, the adopted channel model is employed with density evolution algorithm to obtain good degree distribution pairs for the LDPC code. Thereafter, a modified shortest-path algorithm is used to construct the parity-check matrix for the LDPC code. Simulation results show that the proposed LDPC code performs well in terms of both error rate and data transmission rate.

Multiple Parallel Concatenated Gallager Codes: Code Design and Decoding Techniques

A novel technique to concatenate multiple component low density parity check (LDPC) codes in parallel is presented. Multiple Parallel Concatenated Gallager Codes leverage the advantage of breaking an equivalently long LDPC code into multiple small (lower-complexity) LDPC codes to offer scalability and scope for improving performance in practical implementation for delay-sensitive and resource-constrained applications. The construction as well as two proposed decoding techniques for MPCGCs is evaluated. It is shown that by optimizing the construction of the component LDPC codes and the number of decoding iterations at various channel conditions, the performance of MPCGCs can be significantly improved and their complexity reduced.

Performance enhancement of power line communication systems with efficient low density parity-check codes, noise removal, equalization, and chaotic interleaving

Power Line Communication (PLC) offers a convenient and inexpensive medium for high-speed data transmission; however, this technology still faces difficult challenges. In particular, the Power Line (PL) channel suffers from multipath propagation accompanied with frequency-selective fading. This paper presents a method for constructing Low Density Parity-Check (LDPC) codes for Forward Error-Correction (FEC) on PL channel combined with Chaotic Interleaving (CI) and linear equalization to compensate for burst errors and multipath effects. First, the PL channel characteristics are introduced. A new way of noise modeling, which takes into account both background and impulse noises, is presented. Unlike Middletonʼs class A noise model, the proposed model is suitable for PLC channel. Noise bursts last no longer than 0.1 ms and they are characterized by Poisson distribution with an impulse arrival rate of 0⩽λ⩽5×10−3. In addition, we present a new algorithm for detection and removal of the impulse noise, by which a gain of 1.5 dB could be obtained. Concerning the system design, Zimmermannʼs model for PLC channel is considered, and then the proposed LDPC code construction method is presented. A Modified Shortest-Path (MSP) algorithm of low complexity is applied to obtain a code graph of the desired structure. The parity-check matrix of the code is constructed column-by-column subject to bit-degree and girth constraints. Both high rate and high girth codes have been constructed, e.g., code rates of 0.94, 0.95, 0.96, and 0.97. For better decoding of LDPC codes, an accurate estimation of Signal-to-Noise Ratio (SNR) is required. Therefore, we introduce a new method for SNR estimation based on higher-order signal statistics. The proposed method is especially beneficial at low SNRs. Furthermore, to achieve better immunity to burst and impulse noise, a novel interleaving scheme based on chaotic Baker map is proposed. Eventually, transmission of images on PL channel has been carried out using computer simulations. Clear images were reconstructed and the proposed algorithms were shown to perform well over PLC channel.

On the Girth of Quasi-Cyclic Protograph LDPC Codes

In this paper, we study the relationships between the girth of the Tanner graph of a quasi-cyclic (QC) protograph low-density parity-check (LDPC) code, the lifting degree, and the size and the structure of the base graph. As a result, for a given base graph, we derive a lower bound on the lifting degree as a necessary condition for the lifted graph to have a certain girth. This also provides an upper bound on the girth of the family of graphs lifted from a given base graph with a given lifting degree. The upper bounds derived here, which are applicable to both regular and irregular base graphs with no parallel edges, are in some cases more general and in some other cases tighter than the existing bounds. The results presented in this work can be used to design cyclic liftings with relatively small degree and relatively large girth. As an example, we present new QC protograph LDPC code constructions with girth 8 using fully connected base graphs. These constructions provide upper bounds on the lifting degree required for achieving girth 8 using fully connected base graphs.