Non-binary Quasi-cyclic LDPC Codes Research Articles

Structural Analysis of Nonbinary Cyclic and Quasi-Cyclic LDPC Codes with α-Multiplied Parity-Check Matrices

Structural Analysis of Nonbinary Cyclic and Quasi-Cyclic LDPC Codes with α-Multiplied Parity-Check Matrices

An Expurgating-Based Method for Constructing Multi-Rate Non-Binary Quasi-Cyclic LDPC Codes

This letter presents a new method for constructing the multi-rate non-binary quasi-cyclic low-density parity-check codes by expurgating a high-rate mother code. A matrix-theoretic approach is applied to design the parity-check matrices of the proposed codes. The resulting codes possess a quasi-cyclic structure, which is desirable for hardware implementations of the encoder and decoder. The codes can also be encoded directly with parity-check matrices due to the block dual-diagonal structure. Simulation results show that the codes can achieve good performance within a wide range of code rates both in the waterfall region and in the error-floor region.

Finite-Length Algebraic Spatially-Coupled Quasi-Cyclic LDPC Codes

The replicate-and-mask (R&M) construction of finite-length spatially-coupled (SC) LDPC codes is proposed in this paper. The proposed R&M construction generalizes the conventional matrix unwrapping construction and contains it as a special case. The R&M construction of a class of algebraic spatially coupled (SC) quasi-cyclic (QC) LDPC codes over arbitrary finite fields is demonstrated. The girth, rank, and time-varying periodicity of the proposed R&M SC QC LDPC codes are analyzed. The error rate performance of finite-length nonbinary algebraic SC QC LDPC codes is investigated with window decoding. Compared to the conventional unwrapping construction, it is found through numerical simulations that the R&M construction resulted in SC QC LDPC codes with better block error rate performance and lower error floors. With a flooding schedule decoder, it is shown that the proposed R&M algebraic SC QC LDPC codes have better error performance than the corresponding LDPC block codes and random SC codes. The R&M construction of irregular SC QC LDPC codes is demonstrated. It is shown that low-complexity regular puncturing schemes can be deployed on these codes to construct families of rate-compatible irregular SC QC LDPC codes with good performance.

A Matrix-Theoretic Approach to the Construction of Non-Binary Quasi-Cyclic LDPC Codes

This paper presents two simple and very flexible methods for constructing non-binary (NB) quasi-cyclic (QC) LDPC codes. The proposed construction methods have several known ingredients including base array , masking , binary to non-binary replacement , and matrix-dispersion . By proper choice and combination of these ingredients, NB-QC-LDPC codes with excellent performance can be constructed. The constructed codes can be decoded with a reduced-complexity iterative decoding scheme which significantly reduces the hardware implementation complexity.

Structured quasi‐cyclic low‐density parity‐check codes based on cyclotomic cosets

This study considers the construction of four-cycle free quasi-cyclic low-density parity-check (QC-LDPC) codes. These codes are based on the cyclotomic cosets of q modulo n where q is a prime power, n is a prime and (n, q) = 1. If s is the order of q modulo n, then n − 1 = sl for some integer l. Then there are l distinct cyclotomic cosets Cai , 1 ≤ i ≤ l. Concatenation of the l circulant matrices formed using these sets gives an s × sl matrix M co(n, q). In addition, the Kronecker product of the transpose of ∪ i Cai with itself modulo n gives an sl × sl matrix M kr(n, q). Replacing the entries of these matrices with their associated dispersed binary n × n matrices provides binary QC-LDPC codes with girth at least six. Furthermore, for any prime power q′ such that q′ = n + 1 or q′ ≥ 2n, these two matrices can be used to construct q′-ary QC-LDPC codes with girth at least six. These constructions produce large classes of four-cycle free binary and non-binary QC-LDPC codes. Among the structured LDPC codes, the introduced technique, from code-construction-complexity perspective, which is an engineering factor, is at least among the best ones if not the best. Performance results are presented which show that the codes obtained perform well over additive white Gaussian noise channel with the iterative sum-product decoding algorithm.

Efficient Shuffled Decoder Architecture for Nonbinary Quasi-Cyclic LDPC Codes

In this brief, a shuffled schedule (SS) of the min-max decoding algorithm is proposed for nonbinary low-density parity-check (LDPC) codes. To increase the throughput and reduce the memory requirement, a modified SS (MSS) with much simpler check node processing is also proposed, based on a new shuffled merge algorithm. Numerical simulations for three LDPC codes with different lengths and rates over GF(32) show: 1) both SS and MSS converge faster and have slightly better error performance than the flooding schedule, and 2) the degradation of the MSS in error performance as well as convergence rate is negligible. Finally, an efficient decoder architecture based on the MSS is proposed for quasi-cyclic LDPC codes. The proposed decoder architecture further enhances the decoding throughput with improved check and variable node processing units. The implementation results of an (837, 726) LDPC decoder over GF(32) demonstrate that the proposed architecture outperforms those in previous works.

On the Irregular Nonbinary QC-LDPC-Coded Hybrid Multidimensional OSCD-Modulation Enabling Beyond 100 Tb/s Optical Transport

As a solution to limited bandwidth and high energy consumption of information infrastructure as well as heterogeneity of optical networks, a hybrid multidimensional coded modulation scheme is proposed employing all available electrical and optical degrees of freedom. Electrical basis functions are based on either modified orthogonal polynomials or prolate spheroidal wave functions, while optical ones on polarization and spatial mode states. The proposed scheme is both multi-Tb/s and beyond 100 Tb/s enabling technology. Additionally, the proposed scheme provides the adaptive, software-defined, and dynamic allocation of bandwidth with fine granularity. The adaptive coding is based on irregular nonbinary quasi-cyclic LDPC codes, providing record net coding gains.

A New Nonbinary Quasi-cyclic LDPC Codes Construction Algorithm

An algorithm for constructing nonbinary Quasi-cyclic LDPC (QC-LDPC) codes on the basis of masking and puncturing techniques in consideration of the shortage of algebraic construction algorithm and time-varying performance of wireless channels is proposed . This scheme can reduce the complexity of the hardware implementation, enhance the error correction performance of the LDPC codes constructed by structure algorithm based on masking technology, and improve the average effectiveness of the communication system by realizing a rate-compatible encoding based on puncturing technology. The simulation results show that the nonbinary LDPC codes constructed with masking and puncturing techniques can provide significantly stronger error correction capabilities as compared with those constructed with algebraic method which are used directly.

Computing the Minimum Distance of Nonbinary LDPC Codes

Finding the minimum distance of low-density-parity-check (LDPC) codes is an NP-hard problem. Different from all existing works that focus on binary LDPC codes, we in this paper aim to compute the minimum distance of nonbinary LDPC codes, motivated by the fact that operating in a large Galois field provides one important degree of freedom to achieve both good waterfall and error-floor performance. Our method is based on the existing nearest nonzero codeword search (NNCS) method, but several modifications are incorporated for nonbinary LDPC codes, including the modified error impulse pattern, the dithering method, and the nonbinary decoder. Numerical results on the estimated minimum distances show that a code's minimum distance can be increased by careful selection of nonzero elements of the parity check matrix, or by increasing the mean column weight, or by increasing the size of the Galois field. These results support observations that have been made based on simulated performance in the literature. Finally, we provide an upper bound on the minimum distance for nonbinary quasi-cyclic LDPC codes.

A Network-Efficient Nonbinary QC-LDPC Decoder Architecture

Nonbinary low-density parity-check (LDPC) codes are of great interest due to their better performance cover binary ones when the code length is moderate. However, the cost of decoder implementation for these LDPC codes is still very high. In this paper, a low-complexity VLSI architecture for nonbinary LDPC decoders is presented. By exploiting the intrinsic shifting and symmetry properties of nonbinary quasi-cyclic LDPC (QC-LDPC) codes, significant reduction of memory size and routing complexity can be achieved. These unique features lead to two network-efficient decoder architectures for Class-I and Class-II nonbinary QC-LDPC codes, respectively. Comparison results with the state-of-the-art designs show that for the code example of the 64-ary (1260, 630) rate-0.5 Class-I code, the proposed scheme can save up to 70.6% hardware required by switch network, which demonstrates the efficiency of the proposed technique. The proposed design for the 32-ary (992, 496) rate-0.5 Class-II code can achieve a 93.8% switch network complexity reduction compared with conventional approaches. Furthermore, with the help of a generator for possible solution sequences, both forward and backward steps can be eliminated to offer processing convenience of check node unit (CNU) blocks. Results show the proposed 32-ary (992, 496) rate-0.5 Class-II decoder can achieve 4.47 Mb/s decoding throughput at a clock speed of 150 MHz.

An Efficient Layered Decoding Architecture for Nonbinary QC-LDPC Codes

Compared to binary low-density parity-check (LDPC) codes, nonbinary LDPC codes have better error performance when the code length is moderate. This paper presents an efficient layered decoder architecture for nonbinary quasi-cyclic (QC) LDPC codes using the proposed barrel-shifter-based permutation network and minimum value filter which is used to determine the first few smallest values from a given set. Through the permutation network, the decoding operations related to the multiplications over finite fields can be efficiently handled in the check-node operations, which simplifies the permutations in the variable-node operations and, hence, enables the layered decoder to be realized efficiently. In order to increase the throughput, we utilize the proposed permutation network and the minimum value filter to devise a selective-input min-max decoder architecture. Using a 90-nm CMOS process, we implemented three nonbinary decoders to demonstrate the proposed techniques.

Structured Nonbinary Rate-Compatible Low-Density Parity-Check Codes

While existing works on rate-compatible low-density parity-check (RC-LDPC) codes, either binary or nonbinary, all focus on random-like construction, we in this letter present a novel method to construct structured nonbinary RC-LDPC codes. Protograph-based design with structured puncturing is applied while numerical simulation of code performance is adopted for optimization. The resultant codes are qualified as nonbinary quasi-cyclic LDPC (QC-LDPC) codes which are amenable to high-speed implementation of encoder/decoder. Numerical results show that the proposed codes achieve very good performance.

Large-Girth Nonbinary QC-LDPC Codes of Various Lengths

In this paper, we construct nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes whose parity check matrices consist of an array of square sub-matrices which are either zero matrices or circulant permutation matrices. We propose a novel method to design the shift offset values of the circulant permutation sub-matrices, so that the code length can vary while maintaining a large girth. Extensive Monte Carlo simulations demonstrate that the obtained codes of a wide range of rates (from 1/2 to 8/9) with length from 1000 to 10000 bits have very good performance over both AWGN and Rayleigh fading channels. Furthermore, the proposed method is extended to design multiple nonbinary QC-LDPC codes simultaneously where each individual code can achieve large girth with variable lengths. The proposed codes are appealing to practical adaptive systems where the block length and code rate need to be adaptively adjusted depending on traffic characteristics and channel conditions.

Design of non-binary quasi-cyclic LDPC codes based on multiplicative groups and Euclidean geometries

This paper presents an approach to the construction of non-binary quasi-cyclic (QC) low-density parity-check (LDPC) codes based on multiplicative groups over one Galois field GF(q) and Euclidean geometries over another Galois field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> ). Codes of this class are shown to be regular with girth 6 ≤ g ≤ 18 and have low densities. Finally, simulation results show that the proposed codes perform very well with the iterative decoding.

Efficient Decoder Design for Nonbinary Quasicyclic LDPC Codes

This paper addresses decoder design for nonbinary quasicyclic low-density parity-check (QC-LDPC) codes. First, a novel decoding algorithm is proposed to eliminate the multiplications over Galois field for check node processing. Then, a partially parallel architecture for check node processing units and an optimized architecture for variable node processing units are developed based on the new decoding algorithm. Thereafter, an efficient decoder structure dedicated to a promising class of high-performance nonbinary QC-LDPC codes is presented for the first time. Moreover, an ASIC implementation for a (620, 310) nonbinary QC-LDPC code decoder over GF(32) is designed to demonstrate the efficiency of the presented techniques.

Polarization-multiplexed rate-adaptive non-binary-quasi-cyclic-LDPC-coded multilevel modulation with coherent detection for optical transport networks

In order to achieve high-speed transmission over optical transport networks (OTNs) and maximize its throughput, we propose using a rate-adaptive polarization-multiplexed coded multilevel modulation with coherent detection based on component non-binary quasi-cyclic (QC) LDPC codes. Compared to prior-art bit-interleaved LDPC-coded modulation (BI-LDPC-CM) scheme, the proposed non-binary LDPC-coded modulation (NB-LDPC-CM) scheme not only reduces latency due to symbol- instead of bit-level processing but also provides either impressive reduction in computational complexity or striking improvements in coding gain depending on the constellation size. As the paper presents, compared to its prior-art binary counterpart, the proposed NB-LDPC-CM scheme addresses the needs of future OTNs, which are achieving the target BER performance and providing maximum possible throughput both over the entire lifetime of the OTN, better.

Construction of non-binary quasi-cyclic LDPC codes by arrays and array dispersions - [transactions papers

This paper presents two algebraic methods for constructing high performance and efficiently encodable nonbinary quasi-cyclic LDPC codes based on arrays of special circulant permutation matrices and multi-fold array dispersions. Codes constructed based on these methods perform well over the AWGN and other types of channels with iterative decoding based on belief-propagation. Experimental results show that over the AWGN channel, these non-binary quasi-cyclic LDPC codes significantly outperform Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard-decision Berlekamp-Massey algorithm or algebraic soft-decision Kötter- Vardy algorithm. Also presented in this paper is a class of asymptotically optimal LDPC codes for correcting bursts of erasures. Codes constructed also perform well over flat fading channels. Non-binary quasi-cyclic LDPC codes have a great potential to replace Reed-Solomon codes in some applications in communication environments and storage systems for combating mixed types of noises and interferences.

High Performance Non-Binary Quasi-Cyclic LDPC Codes on Euclidean Geometries LDPC Codes on Euclidean Geometries

This paper presents algebraic methods for constructing high performance and efficiently encodable non-binary quasi-cyclic LDPC codes based on flats of finite Euclidean geometries and array masking. Codes constructed based on these methods perform very well over the AWGN channel. With iterative decoding using a fast Fourier transform based sum-product algorithm, they achieve significantly large coding gains over Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard-decision Berlekamp-Massey algorithm or algebraic soft-decision Kotter-Vardy algorithm. Due to their quasi-cyclic structure, these non-binary LDPC codes on Euclidean geometries can be encoded using simple shift-registers with linear complexity. Structured non-binary LDPC codes have a great potential to replace Reed-Solomon codes for some applications in either communication or storage systems for combating mixed types of noise and interferences.

A unified approach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fields

A unified approach for constructing binary and nonbinary quasi-cyclic LDPC codes under a single framework is presented. Six classes of binary and nonbinary quasi-cyclic LDPC codes are constructed based on primitive elements, additive subgroups, and cyclic subgroups of finite fields. Numerical results show that the codes constructed perform well over the AWGN channel with iterative decoding.

Transactions Papers - Constructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach

This paper is concerned with construction of efficiently encodable nonbinary quasi-cyclic LDPC codes based on finite fields. Four classes of nonbinary quasi-cyclic LDPC codes are constructed. Experimental results show that codes constructed perform well with iterative decoding using a fast Fourier transform based q-ary sum-product algorithm and they achieve significant coding gains over Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard- decision Berlekamp-Massey algorithm or algebraic soft-decision Kotter-Vardy algorithm.