Outer Low-density Parity-check Code Research Articles

Concatenated LDPC/2-D-Marker Codes and Non-Iterative Detection/Decoding for Recovering Position Errors in Racetrack Memories

Racetrack memories (RMs) are prone to alignment faults called position errors (PEs), which manifest as insertions and deletions of stored data bits. Conventional coding schemes for PEs demonstrate the promising results by employing low-density parity-check (LDPC) codes with an iterative detection and decoding algorithm. However, the computational complexity of detection is relatively high. In this article, for channels with PEs, we present a new coding scheme that can effectively decode corrupted data, even if the detection is executed only once (i.e., a non-iterative detection and decoding scenario). The proposed code consists of a concatenation of an inner 2-D marker code, which is specialized for PEs to mitigate the effect of insertion and deletion (ID) errors, and an outer irregular LDPC code. We also provide tractable design methodologies for these constituent codes. First, we identify the 2-D-marker code structures that offer higher achievable information rates in a non-iterative scenario and then optimize irregular LDPC codes to ensure good decoding properties. Through asymptotic-performance analysis and finite-length simulation, we confirm the effectiveness of the proposed coding scheme. Ultimately, the proposed coding scheme has the capability to reduce code rate loss and to provide excellent decoding performance under a non-iterative scenario, which also helps in understanding the reliability of RM when a low-complexity decoding algorithm is used to correct ID errors caused by PEs.

Coding for Two-User Energy Harvesting Interference Channel

A two-user interference channel with energy harvesting transmitters, each equipped with a finite battery, is considered. Achievable rate regions (ARRs) considering independent and identically distributed Shannon strategies at both users and ignoring the memory in the battery state are obtained for both single-user decoding and joint decoding at the receivers. Explicit and implementable codes based on concatenation of a nonlinear trellis code (NLTC) with an outer low-density parity-check code are designed, and it is demonstrated that rate pairs close to the boundary of ARR can be obtained with this approach. Furthermore, an improved alternative decoding scheme which exploits the memory in the battery state is developed, and it is shown to be highly superior to the simple decoding approach via numerical examples. Superiority of the newly developed practical channel coding solutions over the previously known alternative approaches are illustrated via extensive set of examples as well.

Iterative Decoding of LDPC-Based Product Codes and FPGA-Based Performance Evaluation

Low-density parity-check (LDPC) codes have the potential for applications in future high throughput optical communications due to their significant error correction capability and the parallel decoding. However, they are not able to satisfy the very low bit error rate (BER) requirement due to the error floor phenomenon. In this paper, we propose a low-complexity iterative decoding scheme for product codes consisting of very high rate outer codes and LDPC codes. The outer codes aim at eliminating the residual error floor of LDPC codes with quite low implementation costs. Furthermore, considering the long simulation time of computer simulation for evaluating very low BER, the hardware platform is built to accelerate the evaluation of the proposed iterative decoding methods. Simultaneously, the fixed-point effects of the decoding algorithms are also be evaluated. The experimental results show that the iterative decoding of the product codes can achieve a quite low bit error rate. The evaluation using field programmable gate array (FPGA) also proves that product codes with LDPC codes and high-rate algebraic codes can achieve a good trade-off between complexity and throughput.

LLR processing of polar codes in concatenation systems

This paper studies the decoding performance of low-density parity-check (LDPC) codes in a serial concatenation system with polar codes employing the successive cancellation (SC) decoding. It is known that the absolute incorrect log-likelihood ratio (LLR) values from the SC decoding can be very large. This phenomenon dramatically deteriorates the error correcting performance of the outer LDPC codes. In this paper, the LLR values of polar codes are regulated by a log processing before being sent to the LDPC decoder. Simulation results show that the log processing is an efficient approach with a low optimization complexity compared with the existing procedures to improve the performance of the serial concatenation systems.

Low-Complexity Generalized Spatial Modulation Schemes Using Codebook-Assisted MIMO Detectors

Generalized spatial modulation (GSM) is an attractive transmission technique for multiple-input multiple-output systems because of its spectral efficiency and energy efficiency. The maximum likelihood detector provides the optimal performance for GSM signal detection, but also leads to a high level of detection complexity. In this paper, both hard-decision and soft-decision detectors, which are based on a low-complexity codebook-assisted tree-search algorithm, are investigated. In addition, outer low-density parity-check codes are designed such that the concatenated schemes are able to provide error performances, which are very close to those of the schemes using conventional detectors, but with a much lower level of computational complexity.

Code Design for Discrete Memoryless Interference Channels

We study the design of explicit and implementable codes for the two-user discrete memoryless interference channels (DMICs). We consider Han–Kobayashi (HK) type encoding where both public and private messages are used and propose coding techniques utilizing a serial concatenation of a nonlinear trellis code (NLTC) with an outer low-density parity-check (LDPC) code. Since exact analytical treatment of the BCJR decoder for the inner trellis-based code appears infeasible, we analytically investigate the iterative decoding process in the asymptotic regime where the probability of decoding error tends to zero. Based on this approximate analysis, we derive a stability condition for this type of a concatenated coding scheme for the first time in the literature. Furthermore, we use an extrinsic information transfer analysis to design the outer LDPC code while fixing the inner NLTC, and utilize the derived stability condition to accelerate the design process and to avoid code ensembles that potentially produce high error floors. Via numerical examples, we demonstrate that our designed codes achieve rate pairs close the optimal boundary of the HK subregion, which cannot be obtained without the use of nonlinear codes. Also, we verify that the estimated thresholds of the designed codes via finite block length simulations and show that our designs significantly outperform the point-to-point optimal codes, hence demonstrating the need for designs specifically tailored for DMICs.

On Code Design for Joint Energy and Information Transfer

Harvesting energy from radio frequency signals along with transmitting data through them is appealing for different wireless communication scenarios, such as radio frequency identification (RFID) systems and implantable devices. In this paper, we propose a technique to design nonlinear codes for the use in such systems taking into account both energy transmission and error rate requirements. In particular, we propose using concatenation of a nonlinear trellis code (NLTC) with an outer low-density parity-check (LDPC) code. We design the NLTC based on maximization of its free distance. We give necessary and sufficient conditions for its catastrophicity; in order to avoid catastrophic codes, we connect each designed NLTC to a corresponding linear convolutional code allowing for the use of simpler conditions for verification. Furthermore, we use EXIT charts to design the outer LDPC code while fixing the inner NLTC. Via examples, we demonstrate that our designed codes operate at $\sim 0.8$ dB away from the information theoretic limits, and they outperform both regular LDPC codes and optimized irregular LDPC codes for additive white Gaussian noise (AWGN) channels. In addition, we show that the proposed scheme outperforms the reference schemes of concatenating LDPC codes with nonlinear memoryless mappers and using classical linear block codes in a time switching mode.

Asymptotic bounds on the decoding error probability for two ensembles of LDPC codes

Two ensembles of low-density parity-check (LDPC) codes with low-complexity decoding algorithms are considered. The first ensemble consists of generalized LDPC codes, and the second consists of concatenated codes with an outer LDPC code. Error exponent lower bounds for these ensembles under the corresponding low-complexity decoding algorithms are compared. A modification of the decoding algorithm of a generalized LDPC code with a special construction is proposed. The error exponent lower bound for the modified decoding algorithm is obtained. Finally, numerical results for the considered error exponent lower bounds are presented and analyzed.

Optimized LDPC differential-encoded 16QAM scheme for high-speed transmission systems

The 16-ary quadrature amplitude modulation (16QAM) is a high spectral efficient scheme for high-speed transmission systems. To remove the phase ambiguity in the coherent detection system, differential-encoded 16QAM (DE-16QAM) is usually used, however, it will cause performance degradation about 3 dB as compared to the conventional 16QAM. To overcome the performance loss, a serial concatenated system with outer low density parity check (LDPC) codes and inner DE-16QAM is proposed. At the receiver, joint iterative differential demodulation and decoding (ID) is carried out to approach the maximum likelihood performance. Moreover, a genetic evolution algorithm based on the extrinsic information transfer chart is proposed to optimize the degree distribution of the outer LDPC codes. Both theoretical analyses and simulation results indicate that this algorithm not only compensates the performance loss, but also obtains a significant performance gain, which is up to 1 dB as compared to the conventional non-DE-16QAM.

A Constrained-Coding Alternative to MPPM

Multipulse pulse position modulation (MPPM) has been widely proposed to improve data rate over traditional pulse position modulation (PPM) in free-space optical communication systems. Encoders for MPPM are typically lookup-table-based, thus restricting the practical size of MPPM codebooks. Power-of-two-sized MPPM codebooks typically do not have an efficient soft-in soft-out decoder, making them poorly suited for concatenation with an outer code under iterative decoding. In this paper, a new coding technique based on constrained coding is introduced that allows construction of codes which have an efficient encoding algorithm. More importantly, these new codes are suitable for iterative soft-decision decoding in concatenation with an outer error-correcting code. Simulation results for both the Gaussian and Poisson channels show that a serial concatenation with an outer low-density parity-check code (LDPC) can achieve between 2 to 3 dB coding gain over comparable Reed-Solomon and LDPC-coded MPPM systems.

Symbol-Level Synchronization and LDPC Code Design for Insertion/Deletion Channels

We investigate a promising coding scheme over channels impaired by insertion, deletion, and substitution errors, i.e., interleaved concatenation of an outer low-density parity-check (LDPC) code with error-correction capabilities and an inner marker code for synchronization purposes. To limit the decoding latency, we start with a single-pass decoding algorithm, that is, marker code-based synchronization is performed only once per received packet and iterative decoding with information exchange between the inner decoder and outer decoder is not allowed. Through numerical evaluations, we first find the marker code structures which offer the ultimate achievable rate when standard bit-level synchronization is performed. Then, to exploit the correlations in the likelihoods corresponding to different transmitted bits, we introduce a novel symbol-level synchronization algorithm that works on groups of consecutive bits, and show how it improves the achievable rate along with the error rate performance by capturing part of the rate loss due to interleaving. When decoding latency is not an issue and multiple-pass decoding is performed, we utilize extrinsic information transfer (EXIT) charts to analyze the convergence behavior of the receiver, which leads to design of outer LDPC codes with good degree distributions. Finally, design examples are provided along with simulation results which confirm the advantage of the newly designed codes over the ones optimized for the standard additive white Gaussian noise (AWGN) channels, especially for channels with severe synchronization problems.

Reverse Concatenated Coded Modulation for High-Speed Optical Communication

We propose the reverse concatenated code as a forward error correction (FEC) scheme suitable for beyond 100-Gb/s optical transmission. In this scheme, BCH code is used as inner code and low-density parity-check (LDPC) code as outer code. The BCH decoder is implemented based on maximum a posteriori (MAP) decoding such as the BCJR/Ashikmin's algorithm, and an LDPC decoder is based on a min-sum-with-correction-term algorithm. Because maximum a posteriori (MAP) decoding is used as the inner decoder, it provides high accuracy reliabilities to be used in LDPC decoding. We show that proposed FEC scheme performs comparably with much longer LDPC codes of girth 12 for a smaller number of LDPC decoder iterations. Because the outer LDPC code is of medium length and the number of required iterations is low, the proposed concatenated scheme represents an interesting candidate to be used in beyond 100-Gb/s optical transmission. The net coding gain (NCG) of concatenated LDPC(16935,14819)-BCH(64,57) code is 9.62 dB at a bit error rate (BER) of 10-9, whereas the expected NCG at BER of 10-13 is 11.38 dB. This concatenated code outperforms the corresponding turbo-product counterpart with a Chase II decoding algorithm by 0.94 dB at BER of 10-9.

Performance Evaluation of the Reed Solomon and Low Density Parity Check Codes for Blu-ray Disk Channels

In this paper, we evaluate the performances of the Reed–Solomon (RS) and low density parity check (LDPC) codes in order to propose an efficient coding system for the outer code in high density optical recording channels. Hard decision decoding (HDD) RS code, iterative soft decision decoding (SDD) RS code or LDPC code is used as the outer code and LDPC code is used as the inner code. We compare the bit error rate (BER) performance of each coding system for various code rates, code lengths and burst errors. Simulation results show that the LDPC code has a better BER performance than the HDD RS and iterative SDD RS codes as the outer code in high density optical channels. Therefore, the LDPC code could be more attractive as the outer code than the RS code in high density optical channels.

Capacity-Achieving Codes With Bounded Graphical Complexity and Maximum Likelihood Decoding

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, the existence of capacity-achieving codes for memoryless binary-input output-symmetric (MBIOS) channels under maximum-likelihood (ML) decoding with bounded graphical complexity is investigated. Graphical complexity of a code is defined as the number of edges in the graphical representation of the code per information bit and is proportional to the decoding complexity per information bit per iteration under iterative decoding. Irregular repeat-accumulate (IRA) codes are studied first. Utilizing the asymptotic average weight distribution (AAWD) of these codes and invoking Divsalar's bound on the binary-input additive white Gaussian noise (BIAWGN) channel, it is shown that simple nonsystematic IRA ensembles outperform systematic IRA and regular low-density parity-check (LDPC) ensembles with the same graphical complexity, and are at most 0.124 dB away from the Shannon limit. However, a conclusive result as to whether these nonsystematic IRA codes can really achieve capacity cannot be reached. Motivated by this inconclusive result, a new family of codes is proposed, called low-density parity-check and generator matrix (LDPC-GM) codes, which are serially concatenated codes with an outer LDPC code and an inner low-density generator matrix (LDGM) code. It is shown that these codes can achieve capacity on any MBIOS channel using ML decoding and also achieve capacity on any BEC using belief propagation (BP) decoding, both with bounded graphical complexity. Moreover, it is shown that, under certain conditions, these capacity-achieving codes have linearly increasing minimum distances and achieve the asymptotic Gilbert–Varshamov bound for all rates. </para>

A Coded Modulation Scheme for Deep-Space Optical Communications

A coded modulation scheme for deep-space optical communications is studied, which is a serial concatenation of an outer low-density parity-check (LDPC) code, an interleaver, a bit-accumulator, and pulse-position modulation (PPM). It is referred as LDPC-APPM. It is decoded with an iterative demodulator-decoder using standard turbo-decoding techniques. Simulation results show that the LDPC-APPM with an outer LDPC code of medium code length suffers a loss of about 0.7 dB at a bit-error rate of relative to the serially concatenated PPM (SCPPM) of similar parameters, which consists of the serial concatenation of an outer convolutional code, an interleaver, a bit-accumulator, and PPM; however, it has better code rate flexibility especially for high rates and simpler implementation structure than SCPPM. Further, it can achieve better performance than the LDPC-PPM of the same rate and length, which consists of the serial concatenation of an LDPC code and PPM.

Serial concatenation of LDPC codes and differential modulations

In this paper, we consider serially concatenated schemes with outer novel and efficient low-density parity-check (LDPC) codes and inner modulations effective against channel impairments. With a pragmatic approach, we show how to design LDPC codes tailored for simple and robust modulation formats, such as differentially encoded (DE) modulations. The LDPC codes are optimized through the use of a recently proposed analysis technique based on extrinsic information transfer (EXIT) charts. In particular, we optimize, through a clever random walk in the parametric space, the degree distributions of the outer LDPC codes, obtaining significant insights on the impact of such distributions on the performance of the proposed concatenated schemes. The optimization is carried out for transmission over both the additive white Gaussian noise channel and a noncoherent channel. The performance predicted by the EXIT chart-based optimization is confirmed by results obtained via computer simulations, considering phase-shift keying and quadrature amplitude modulation at the transmitter side, and iterative demodulation/decoding at the receiver side. The significance of the proposed optimized design of LDPC-coded schemes with DE modulations is validated by the fact that standard nonoptimized LDPC codes perform poorly when used together with inner DE modulations.

Matched Information Rate Codes for Partial Response Channels

In this paper, we design capacity-approaching codes for partial response channels. The codes are constructed as concatenations of inner trellis codes and outer low-density parity- check (LDPC) codes. Unlike previous constructions of trellis codes for partial response channels, we disregard any algebraic properties (e.g., the minimum distance or the run-length limit) in our design of the trellis code. Our design is purely probabilistic in that we construct the inner trellis code to mimic the transition probabilities of a Markov process that achieves a high (capacity-approaching) information rate. Hence, we name it a matched information rate (MIR) design. We provide a set of five design rules for constructions of capacity-approaching MIR inner trellis codes. We optimize the outer LDPC code using density evolution tools specially modified to fit the superchannel consisting of the inner MIR trellis code concatenated with the partial response channel. Using this strategy, we design degree sequences of irregular LDPC codes whose noise tolerance thresholds are only fractions of a decibel away from the capacity. Examples of code constructions are shown for channels both with and without spectral nulls.