Repeat-accumulate Codes Research Articles

Design of Rate-Compatible Anytime Codes Based on Spatially Coupled Repeat-Accumulate Codes

Design of Rate-Compatible Anytime Codes Based on Spatially Coupled Repeat-Accumulate Codes

Joint Source Channel Anytime Coding Based on Spatially Coupled Repeat-Accumulate Codes

Joint Source Channel Anytime Coding Based on Spatially Coupled Repeat-Accumulate Codes

Robust Rateless Spatially Coupled Repeat-Accumulate-Repeat Multi-User Codes on IDMA Systems

In this paper, we design a kind of robust rateless multi-user code, which is called spatially coupled repeat-accumulate-repeat (SC-RA-R) code, for the interleave-division multiple-access (IDMA) system. The code is constructed by serially concatenating a fixed outer spatially coupled repeat-accumulate (SC-RA) code with an adjustable inner irregular repetition code. It achieves rateless property by simply adjusting the parameters of the inner repetition code. The IDMA system with the proposed SC-RA-R multi-user codes can adapt to varying channel qualities and varying number of users via rateless property while keeping encoder and decoder implementation unchanged. Analytical extrinsic information transfer (EXIT) functions are derived to determine belief propagation (BP) decoding thresholds of SC-RA-R codes on the IDMA system. The optimization expression is also given under the constraint that repeat parameters are monotonically non-decreasing with the sum rate decreasing. The numerical results show that the maximum achievable sum rates of the SC-RA-R codes can be close to the Shannon bound for arbitrary channel qualities. Besides, our proposed SC-RA-R codes have better BP thresholds and bit-error-rate (BER) performances than the conventional optimal parallel-concatenate codes (PCC) and the optimal partially repeated spatially coupled low-density parity-check (PR-SC-LDPC) codes on IDMA systems.

Log-Logarithmic Time Pruned Polar Coding

A pruned variant of polar coding is proposed for binary erasure channels. For sufficiently small $\varepsilon>0$, we construct a series of capacity achieving codes with block length $N=\varepsilon^{-5}$, code rate $R=\text{Capacity}-\varepsilon$, error probability $P=\varepsilon$, and encoding and decoding time complexity $\text{bC}=O(\log\left|\log\varepsilon\right|)$ per information bit. The given per-bit complexity $\text{bC}$ is log-logarithmic in $N$, in $\text{Capacity}-R$, and in $P$; no known family of codes possesses this property. It is also the second lowest $\text{bC}$ after repeat-accumulate codes and their variants. While random codes and classical polar codes are the only two families of capacity-achieving codes whose $N$, $R$, $P$, and $\text{bC}$ were written down as explicit functions, our construction gives the third family. Then we generalize the result to: Fix a prime $q$ and fix a $q$-ary-input discrete symmetric memoryless channel. For sufficiently small $\varepsilon>0$, we construct a series of capacity achieving codes with block length $N=\varepsilon^{-O(1)}$, code rate $R=\text{Capacity}-\varepsilon$, error probability $P=\varepsilon$, and encoding and decoding time complexity $\text{bC}=O(\log\left|\log\varepsilon\right|)$ per information bit. The later construction gives the fastest family of capacity-achieving codes to date on those channels.

Multisource SWIPT-based coded cooperation:rate compatible codes and codeword splitting protocol

To achieve a high reliable and energy-saving green communication, we investigate a multisource simultaneous wireless information and power transfer (SWIPT)-based coded cooperation where the relay can realize information decoding and energy harvesting. Firstly, a class of naturally rate compatible low-density parity-check (LDPC) codes–quasi-cyclic repeat-accumulate (QC-RA) codes is introduced, and the joint parity-check matrix corresponding to the QC-RA codes employed by the multiple sources and relay is deduced. Based on the joint parity-check matrix, we jointly design the QC-RA codes to cancel all the short girth cycles. Then, by exploiting the rate compatible characteristic of QC-RA codes, we propose a new SWIPT protocol—codeword splitting protocol for the proposed system, which has the characteristics of lower complexity, higher efficiency, no strictly bit synchronization limitation, and less hardware requirement. The results show that the bit error rate (BER) performance of the proposed system employing jointly designed QC-RA codes clearly outperforms that of general RA codes. Theoretical analysis and numerical simulations also demonstrate the superiority of the proposed codeword splitting protocol.

Capacity-Achieving MIMO-NOMA: Iterative LMMSE Detection

This paper considers a low-complexity iterative Linear Minimum Mean Square Error (LMMSE) multi-user detector for the Multiple-Input and Multiple-Output system with Non-Orthogonal Multiple Access (MIMO-NOMA), where multiple single-antenna users simultaneously communicate with a multiple-antenna base station (BS). While LMMSE being a linear detector has a low complexity, it has suboptimal performance in multi-user detection scenario due to the mismatch between LMMSE detection and multi-user decoding. Therefore, in this paper, we provide the matching conditions between the detector and decoders for MIMO-NOMA, which are then used to derive the achievable rate of the iterative detection. We prove that a matched iterative LMMSE detector can achieve (i) the optimal capacity of symmetric MIMO-NOMA with any number of users, (ii) the optimal sum capacity of asymmetric MIMO-NOMA with any number of users, (iii) all the maximal extreme points in the capacity region of asymmetric MIMO-NOMA with any number of users, (iv) all points in the capacity region of two-user and three-user asymmetric MIMO-NOMA systems. In addition, a kind of practical low-complexity error-correcting multiuser code, called irregular repeat-accumulate code, is designed to match the LMMSE detector. Numerical results shows that the bit error rate performance of the proposed iterative LMMSE detection outperforms the state-of-art methods and is within 0.8dB from the associated capacity limit.

Multilevel Coding With Spatially Coupled Repeat-Accumulate Codes for High-Order QAM Optical Transmission

We propose multilevel coding (MLC) with spatially coupled codes to increase the net coding gain (NCG) and reduce the power consumption of forward error correction (FEC) codes in high-order quadratic-amplitude modulation (QAM) optical transmissions. We optimized the degree distribution of the spatially coupled repeat-accumulate (SC-RA) codes by using density evolution to enhance the effect of spatial coupling with a limited code length. We constructed multilevel codes with the SC-RA codes and investigated their performance in numerical simulations. NCGs of 12.5, 13.2, and 13.7 dB were obtained for 16, 64, and 256 QAM. We found that MLC is more efficient in terms of circuit size and computational complexity than the conventional FEC configuration of bit-interleaved coded modulation (BICM). Moreover, its efficiency tends to increase as the modulation order increases. Our estimates suggest that the power consumed in the FEC process could be reduced to about half for 16 QAM, 35% for 64 QAM, and 15% for 256 QAM compared with when BICM is used to obtain similar NCG values.

Mapping Design for $2^{M}$ -Ary Bit-Interleaved Polar Coded Modulation

This paper proposes a mapping design for bit-interleaved polar coded modulation (BIPCM) systems with belief propagation (BP) decoding. We first introduce a two-layer bipartite graph to represent BIPCM, where a new mapping graph linking polar graph to modulator is added to the conventional factor graph. Then, a mapping design is proposed and the design paradigm is to separate sub-channels with lower reliability to different stopping trees of polar codes, aiming to make sure that each stopping tree receives reliable extrinsic information from demodulator. The proposed mapping algorithm is employed for BIPCM with traditional polar codes over 16-quadrature amplitude modulation (QAM) and 256-QAM. Numerical results show that our scheme can improve the error-correcting performance compared to the conventional scheme with a random mapping. Furthermore, to meet code-length requirement of different modulation orders, we propose an efficient method to construct flexible-length polar code (FLPC) by coupling several short length polar codes with a repeat-accumulate (RA) code. Also, the proposed FLPC is employed in the BIPCM system, with the designed mapping algorithm, simulation result also reveals that the block error rate performance of proposed BIPCM scheme with BP decoding outperforms the one with successive cancellation decoding by providing a gain up to 1 dB.

High-Speed Reconciliation for CVQKD Based on Spatially Coupled LDPC Codes

The speed of continuous variable quantum key distribution is limited by the reconciliation efficiency and the reconciliation frame error rate (FER) in the reconciliation phase. In this paper, we propose an approach that may increase the reconciliation efficiency and decrease the FER. In detail, to increase the reconciliation efficiency, rather than using the block low-density parity-check (LDPC) codes, such as quasi-cyclic LDPC codes, multiedge-type LDPC codes, and punctured LDPC codes, this paper introduces spatially coupled (SC) LDPC codes to the reconciliation phase. To decrease the FER, we propose a new reconciliation scheme based on the special structure of SC-LDPC codes. To demonstrate the performance of the proposed approach, we construct the SC-LDPC codes based on quasi-cyclic repeat-accumulate codes. It is shown that the proposed scheme leads to higher reconciliation speed than that of the previous reconciliation schemes.

On the Design of Multi-Dimensional Irregular Repeat-Accumulate Lattice Codes

Most multi-dimensional (more than two dimensions) lattice partitions only form additive quotient groups and lack multiplication operations. This prevents us from constructing lattice codes based on multi-dimensional lattice partitions directly from non-binary linear codes over finite fields. In this paper, we design lattice codes from Construction A lattices where the underlying linear codes are non-binary irregular repeat-accumulate (IRA) codes. Most importantly, our codes are based on multi-dimensional lattice partitions with finite constellations . We propose a novel encoding structure that adds randomly generated lattice sequences to the encoder’s messages, instead of multiplying lattice sequences to the encoder’s messages. We prove that our approach can ensure that the decoder’s messages exhibit permutation-invariance and symmetry properties. With these two properties, the densities of the messages in the iterative decoder can be modeled by Gaussian distributions described by a single parameter. With Gaussian approximation, extrinsic information transfer charts for our multi-dimensional IRA lattice codes are developed and used for analyzing the convergence behavior and optimizing the decoding thresholds. Simulation results show that our codes can approach the unrestricted Shannon limit within 0.46 dB and outperform the previously designed lattice codes with 2-D lattice partitions and existing lattice coding schemes for large codeword length.

Design Guidelines of Low-Density Parity-Check Codes for Magnetic Recording Systems

As one of the most classical data-storage systems, magnetic recording (MR) systems have attracted a significant amount of research attention in the past several decades due to the advantages of low cost and high storage density. Along with the continual increase of areal density of modern MR devices, more severe inter-symbol interference (ISI) and noise appear and thus reliable storage becomes more difficult. To address this challenging problem, turbo detections and error-correction codes (ECCs) have been applied to MR systems so as to significantly improve the overall data-storage reliability. Among all the existing ECCs, low-density parity-check (LDPC) codes are of particular interest because they can offer excellent error performance with relatively low encoding and decoding complexity. This paper presents a comprehensive survey of the latest research advancements in LDPC-code design for MR systems from the perspectives of code construction, decoder design, as well as asymptotic performance-evaluation methodology. More specifically, we summarize the design guidelines of LDPC encoder and decoder over both one-dimensional (OD) ISI and two-dimensional (TD) ISI channels, which are commonly used to characterize MR systems with different areal densities. We also concisely portray the research progress in the design of some LDPC-code variants, such as protograph codes, repeat-accumulate codes, spatially coupled codes, and their non-binary counterparts over the aforementioned ISI channels. In particular, we restrict our attention to the reading process and ignore the written-in errors in MR systems unless otherwise stated. Hopefully, this survey will inspire more research activities in the area of LDPC-coded MR systems.

Delay-Universal Channel Coding With Feedback

In this paper, the design of error-correcting or channel codes for delay-universal/anytime communication is shown while considering systems with and without a feedback link. We construct practical and low complexity anytime channel codes based on spatially-coupled repeat-accumulate (SC-RA) codes. Performance and density evolution analysis are shown for the binary erasure channel (BEC) and the binary input additive white Gaussian noise (BIAWGN) channel. We observe that the erasure/error floors exist even at low decoding delay in the following cases: 1) when the code rate is close to the Shannon capacity; and/or 2) when the code parameters are chosen to target a high decaying rate of erasure/error probability. To mitigate erasure/error floors, we present feedback algorithms for BEC and BIAWGN channels. We show that the proposed feedback strategies can greatly enhance the performance of anytime SC-RA codes. Numerical results also show that the feedback strategies significantly reduce the decoder complexity. The proposed feedback approach is applied to an aircraft tracking application to track/calculate/estimate the state information of the aircraft. Based on comparisons of the results obtained from the traditional block and anytime coding scenarios, it is observed that the latter significantly outperforms the former in terms of tracking performance.

A Low-Complexity Multiuser Coding Scheme With Near-Capacity Performance

Nonorthogonal multiuser coding is an essential technique for superimposed transmission and improving spectrum efficiency of future wireless communication networks. In this paper, a low-complexity nonorthogonal coding scheme, called multiuser repetition-aided irregular repeat-accumulate (IRA) code, is proposed for a multiple access channel (MAC), and the scheme can be applied to fifth-generation (5G) nonorthogonal multiple access systems. The main idea is that not only parity checks, which are generated by an IRA encoder, but repetitions as well, are used in each user's codeword to reduce the coding and decoding complexities. Repetition is a simple way to construct a low-rate code and is shown to be beneficial for multiuser decoding iteration. With a deliberate control of the fraction of repetitions in the codeword and a degree distribution optimization of the IRA encoder, near MAC capacity performance is achieved. It is shown that as the number of users increases, more repetitions can be used, and therefore, very low encoding and decoding complexities are required.

Simplified multiuser code design for MIMO–NOMA

Combination of multiple-input-multiple-output and non-orthogonal multiple access (MIMO–NOMA) is a promising multiple-access technology, which can greatly improve spectral efficiency and reduce latency. Among major topics of MIMO–NOMA, an interesting one is how to construct suitable multiuser codes for MIMO–NOMA. In previous works, multiuser codes require to be redesigned when the number of users, transmit antennas, or receive antennas change. Therefore, the previous design methods are too complicated to be applied to MIMO–NOMA. To solve this problem, in this study, the authors first propose a simple multiuser detector (MUD) that detects the signal for each user by regarding the superimposed signal from the other users as interference. Then, based on extrinsic information transfer analysis for the MUD, they propose three criteria to simplify the code design, which copes with the changes of user number and antenna configuration. Moreover, when user number is large, each user requires a low-rate code to overcome the severe multiuser interference. On the basis of the proposed criteria, they design a low-rate repetition-aided irregular repeat-accumulate (Rep-IRA) code for MIMO–NOMA with different numbers of users and antennas, which can achieve low complexity with the aid of repetition.

EXIT Chart-Based IRA Code Design for TDMR Turbo-Equalization System

We present an extrinsic information transfer (EXIT) chart-based design technique for irregular repeat-accumulate (IRA) codes used in 2-D magnetic recording (TDMR) turbo-equalization systems. The channel model includes Voronoi magnetic grains, 2-D intersymbol interference (2D-ISI) and additive white Gaussian noise (AWGN). The receiver uses a 2D-ISI BCJR equalizer and an IRA decoder. For one outer equalizer-decoder iteration, we propose theory and simulation-based methods for computing EXIT curves. The simulation method calculates experimental EXIT curves for the check node decoder (CND) and the combination of the variable node decoder (VND) and an equalizer. The theoretical approach recursively calculates CND and VND Gaussian mixture model parameters in order to calculate EXIT curves. We then fit the VND and CND EXIT curves to find optimized variable node degree distributions. Simulation results show that the TDMR-optimized IRA codes achieve up to a 6.2% density increase in user bits/grain (U/G) compared with IRA codes designed for AWGN channels. The theory-based code designs achieve the same or better U/G as the simulation-based designs, but require 98% less design computation time. We also derive optimized IRA codes for iterative turbo-equalization; these codes can achieve simultaneous U/G gains and SNR savings compared with AWGN-optimized codes.

Optimal rate profile for multi‐user multi‐rate transmission systems by bivariate fixed‐point analysis

A K-user multi-rate code is proposed for a Gaussian multiple access channel with binary inputs, equal-power, and symbol synchronisation. In this multi-rate transmission, K users are equally divided into M groups. For each user in the mth group, a rate- 1 / q m regular repeat-accumulate code serially concatenated with a length- L m spreading is employed. The transmitted rate of each user in the mth group is 1 / ( q m L m ) . At the receiver, iterative joint decoding (IJD) and hybrid interference cancellation (HIC) schemes are considered. For each decoding scheme, a bivariate fixed-point analysis is applied to explicitly represent ( q m , L m ) as a function of mutual information outputs. On the basis of these basic explicit representations, a united unreliable region is given, where users in at least one group are undecodable. The complementary set of the united unreliable region gives an optimal rate profile that achieves the maximum sum rate. Numerical results show that, for the IJD scheme with M increments, the maximum sum rate increases, approaches the Shannon limit, and exceeds that in conventional equal rate transmission. The maximum sum rate of the HIC scheme, which provides much lower decoding complexity than the IJD scheme, is superior to the conventional successive interference cancellation scheme.

Energy-harvesting-based RA-coded cooperative MIMO: Codes design and performance analysis

To overcome the power limitation and prolong the lifetime of the coded cooperation system, this paper proposes an energy-harvesting-based repeat-accumulate (RA)-coded cooperative multiple-input multiple-output (MIMO), where the relay is equipped with multiple antennas for information decoding and energy harvesting. Firstly, a kind of structured low-density parity-check (LDPC) codes – RA codes are introduced, and the RA codes for the source and relay are designed jointly to cancel all girth-4 cycles. Secondly, the optimal antenna selection algorithm is adopted by the relay to transmit the information. Furthermore, the outage probability and bit error rate (BER) of the system are analyzed. Theoretical analysis and numerical simulations show that the investigated system outperforms the corresponding point-to-point system under the same condition. Simulation result also demonstrates that BER performance of the system employing jointly designed RA codes is much better than that of general RA codes.

Partial parallel encoding and algorithmic construction of non-binary structured IRA codes

The non-binary (NB) Irregular Repeat Accumulate (IRA) codes, as a subclass of NB LDPC codes, potentially have an excellent error-correcting performance. They are also known to provide linear complexity of encoding, but the basic encoding method with the serial rate-1 accumulator significantly limits the encoder throughput. Then the objective of the research presented in this paper is to develop an encoding method providing significantly increased throughput of an NB-IRA encoder altogether with a flexible code construction methods for the structured (S-NB-IRA) codes eligible for the proposed encoding method. For this purpose, we reformulate the classic encoding algorithm to fit into the partial parallel encoder architecture. We propose the S-NB-IRA encoder block diagram and show that its estimated throughput is proportional to the submatrix size of the parity check matrix, which guarantees a wide complexity-throughput tradeoff. Then, in order to facilitate the S-NB-IRA coding systems design, we present a computer search algorithm for the construction of good S-NB-IRA codes. The algorithm aims at optimizing the code graph topology along with selecting an appropriate non-binary elements in the parity check matrix. Numerical results show that the constructed S-NB-IRA codes significantly outperform the binary IRA and S-IRA codes, while their performance is similar to the best unstructured NB-LDPC codes.

Continuous Transmission of Spatially Coupled LDPC Code Chains

We propose a novel encoding/transmission scheme called continuous chain (CC) transmission that is able to improve the finite-length performance of a system using spatially-coupled low-density parity-check (SC-LDPC) codes. In CC transmission, instead of transmitting a sequence of independent codewords from a terminated SC-LDPC code chain, we connect multiple chains in a layered format, where encoding, transmission, and decoding are now performed in a continuous fashion. The connections between chains are created at specific points, chosen to improve the finite-length performance of the code structure under iterative decoding. We describe the design of CC schemes for different SC-LDPC code ensembles constructed from protographs: a (J,K)-regular SC-LDPC code chain, a spatially-coupled repeat-accumulate (SC-RA) code, and a spatially-coupled accumulate-repeat-jagged-accumulate (SC- ARJA) code. In all cases, significant performance improvements are reported and, in addition, it is shown that using CC transmission only requires a small increase in decoding complexity and decoding delay with respect to a system employing a single SC-LDPC code chain for transmission.

On the Waterfall Performance of Finite-Length SC-LDPC Codes Constructed From Protographs

An analysis of spatially coupled low-density parity-check (SC-LDPC) codes constructed from protographs is proposed. Given the protograph used to generate the SC-LDPC code ensemble, a set of scaling parameters to characterize the average finite-length performance in the waterfall region is computed. The error performance of structured SC-LDPC code ensembles is shown to follow a scaling law similar to that of unstructured randomly constructed SC-LDPC codes. Under a finite-length perspective, some of the most relevant SC-LDPC protograph structures proposed to date are compared. The analysis reveals significant differences in their finite-length scaling behavior, which is corroborated by simulation. Spatially coupled repeat-accumulate codes present excellent finite-length performance, as they outperform in the waterfall region SC-LDPC codes of the same rate and better asymptotic thresholds.