Iterative Soft-decision Decoding Research Articles

A Scheme for Collective Encoding and Iterative Soft-Decision Decoding of Cyclic Codes of Prime Lengths: Applications to Reed–Solomon, BCH, and Quadratic Residue Codes

A novel scheme is presented for encoding and iterative soft-decision decoding of cyclic codes of prime lengths. The encoding of a cyclic code of a prime length is performed on a collection of codewords which are mapped through Galois Fourier transform into a codeword in a low-density parity-check code with a binary parity-check matrix for transmission. Using this matrix, binary iterative soft-decision decoding algorithm is applied to jointly decode a collection of codewords from the cyclic code. The joint-decoding allows for information sharing among the received vectors corresponding to the codewords in the collection during the iterative decoding process. For decoding Reed-Solomon and BCH codes of prime lengths, the proposed decoding scheme not only requires much lower decoding complexity than other soft-decision decoding algorithms for these codes, but also yields superior performance. The proposed decoding scheme can also achieve a joint-decoding gain over the maximum likelihood decoding of individual codewords. The decoding scheme is also applied to quadratic residue codes.

Baseline parity‐check matrix for iterative soft‐decision decoding of binary cyclic codes

Belief propagation (BP) is a soft-decision (SD) decoder that is commonly used to obtain near-optimal decoding performance for linear codes defined over sparse parity-check matrix. Nevertheless, the BP yields poor performance when used in the decoding of algebraic block codes, usually described by a dense parity-check matrix. Hence, enhanced BP decoders transform the parity check matrix of such codes at each iteration for efficient decoding. In this article, the performance of the transformed parity-check matrix of the iterative SD decoder is analysed for the class of binary cyclic codes using a perfect knowledge model (PKM). The PKM computes a list of candidate matrices and selects a baseline parity-check matrix according to a distance metric. The selected matrix is optimal since it minimizes the probability of error over various choices in the list. Results show that, for a given channel condition, the conventional transformed matrix obtained by Gaussian elimination is sub-optimal and does not necessarily contain unitary weighted columns at corresponding columns of the unreliable bits. Moreover, PKM can be used to verify the performances of newly developed iterative SD decoders for binary cyclic codes based on parity-check equations instead of maximum-likelihood decoding.

A Generalized Parity-Check Transformation for Iterative Soft-Decision Decoding of Binary Cyclic Codes

The conventional parity-check matrix transformation algorithm (PTA) requires matrix inversion to transform matrices of maximum distance separable (MDS) codes. However, such matrix transformation is not always guaranteed for the class of non-MDS codes. Hence, the PTA fails for binary cyclic (BC) codes. To overcome this limitation, we developed a generalized parity-check matrix transformation (GPT) algorithm for binary cyclic codes. The GPT avoids the matrix inversion step of the PTA. It permutes the columns of the parity-check matrix based on the reliability information from the channel. Results show a significant bit error rate (BER) performance gain as compared to the existing PTA. It also presents a reasonable BER performance as compared to the other soft-decision (SD) decoding algorithms. In addition, the decoder functions within a practical decoding time complexity, particularly at the message passing stage.

RBER Aware Multi-Sensing for Improving Read Performance of 3D MLC NAND Flash Memory

3D-multi-level cell (MLC) NAND flash memory adopts 3D stack technology and stores two bits per cell, leading to improved storage capacities, but sacrificing data reliability. Low-density parity-check (LDPC) codes showing strong error correction capability benefit from their soft decision decoding, which is widely exploited to guarantee data reliability. Nevertheless, adopting LDPC codes can introduce a concern about read performance, because their iterative soft-decision decoding requires Log-Likelihood Ratio (LLR) information, called soft decision information, by applying multi-sensing voltages. This process of obtaining LLR information leads to high sensing and transferring latencies, lowering down 3D-MLC read performance. In particular, when raw bit error rates (RBER) are much higher due to the long retention periods and program/erase (P/E) cycles, this problem becomes more serious. In this paper, we propose a RBER-aware multi-sensing scheme for reducing sensing and transferring latencies, and thus improving read performance. This proposed scheme exploits the variations of RBER in flash pages with the increase of retention time and P/E cycles to dynamically apply sensing voltages. Simulation results show that this scheme significantly decreases the number of required sensing voltages while maintaining LDPC error correction capability, enhancing 3D-MLC read performance.

Bandwidth-Efficient Coded Modulation Schemes for Physical-Layer Network Coding with High-Order Modulations

This paper presents several soft decision iterative decoding schemes for physical-layer network coding (PNC) operated with coded modulation (CM) and bit-interleaved coded modulation (BICM). With respect to PNC operated with CM, we consider network coding-based channel decoding (NC-CD) and multi-user complete decoding (MUD-NC) for PNC decoding at the relay. Their BICM counterparts are XOR-based channel decoding (XOR-CD) and MUD-XOR, respectively. First, we show that, when the decoding is non-iterative, there is a gap between the BICM capacities of both XOR-CD and MUD-XOR under Gray mapping and the capacities of their CM counterparts, NC-CD, and MUD-NC. This is in contrast to the conventional point-to-point communication system, for which the BICM capacity with Gray mapping is known to be very close to the CM capacity, without the need for iterative decoding. Second, we investigate the error performance of iteratively decoded BICM XOR-CD and MUD-XOR. Extrinsic information transfer chart analysis and simulation results indicate that for these Gray-mapped BICM PNC systems, iterative decoding can achieve considerable gains over non-iterative decoding. Again, this is in contrast to the Gray-mapped BICM point-to-point communication system, for which iterative decoding provides little gain over non-iterative decoding. We further show that Gray mapping gives rise to best PNC rate for MUD-XOR and XOR-CD systems among several bits-to-symbol mappings under study. Overall, our results indicate that BICM PNC systems exhibit different decoding behavior from conventional BICM point-to-point systems. This paper serves as a first foray into the investigation of this issue.

Symbol level iterative soft decision decoder for Reed‐Solomon codes based on parity‐check equations

A symbol level iterative soft decision (SD) algorithm for Reed-Solomon codes based on parity-check equations is developed. This is achieved by transforming the systematic parity-check matrix according to some rules. The rules are based on the soft reliability information matrix derived from the received vector. The symbol error rate performance of the resulting algorithm is documented through computer simulation and compared with the hard decision Berlekamp-Massey (B-M) algorithm, and the Koetter and Vardy-Guruswami and Sudan (KV-GS) algorithm. The result verifies that the iterative (SD) algorithm outperforms the KV-GS and B-M algorithms by a significant margin while maintaining a reasonable decoding time complexity level.

Terminated and Tailbiting Spatially Coupled Codes With Optimized Bit Mappings for Spectrally Efficient Fiber-Optical Systems

We study the design of spectrally efficient fiber-optical communication systems based on different spatially coupled (SC) forward error correction (FEC) schemes. In particular, we optimize the allocation of the coded bits from the FEC encoder to the modulation bits of the signal constellation. Two SC code classes are considered. The codes in the first class are protograph-based low-density parity-check (LDPC) codes which are decoded using iterative soft-decision decoding. The codes in the second class are generalized LDPC codes which are decoded using iterative hard-decision decoding. For both code classes, the bit allocation is optimized for the terminated and tailbiting SC cases based on a density evolution analysis. An optimized bit allocation can significantly improve the performance of tailbiting SC codes codes over the baseline sequential allocation, up to the point where they have a comparable gap to capacity as their terminated counterparts, at a lower FEC overhead. For the considered terminated SC codes, the optimization only results in marginal performance improvements, suggesting that in this case a sequential allocation is close to optimal.

Efficient Decoding of Short Length Linear Cyclic Codes

Iterative soft decision decoding of linear block codes is a practical necessity when working with even modest block lengths. A number of algorithms have been proposed in the literature which use the permutation group of a code and the belief propagation (BP) algorithm for decoding. A novel soft-input, soft-output algorithm is presented that can be used for efficiently decoding of linear cyclic codes. Utilising the automorphism property of cyclic codes the permutation is incorporated into the belief propagation algorithm resulting in faster convergence and better error correcting performance. Performance of the new approach is analysed using a (63,45) BCH code and a (72,36) quadratic residue code. Simulation results show significant reduction in the average number of required decoding iterations and some improvement in error correcting performance over published algorithms.

Soft decision iterative error and erasure decoder for Reed–Solomon codes

A list decoding algorithm for iterative soft-decision decoding of Reed–Solomon (RS) codes is proposed in this study. The algorithm erases a combination of least reliable symbols and iteratively flips a new combination of least reliable bits in the received sequence. The symbol error correction capability of the algebraic hard decision RS decoder is extended up to the minimum Hamming distance of the code. A gain in performance is exhibited either at a lower or similar iteration complexity as compared to other iterative soft decision decoders of similar type. A complexity comparison with some of the comparable soft decision decoders is given. Simulation results for different modulation schemes and channel types are presented in this study for a comparison with other soft decision decoders of a similar kind.

Adaptive Hybrid ARQ System Using Turbo Product Codes with Hard/Soft Decoding

The bit error rate (BER) performance of turbo product codes (TPC) has been considered extensively in the literature. However, other performance metrics such as throughput can be more informative in particular systems. In this letter, the throughput performance of hybrid automatic repeat request (HARQ) is considered using TPC with iterative hard and soft decision decoding. Monte Carlo simulation and semi-analytical solutions are developed to evaluate the throughput of HARQ-TPC system for a wide selection of codes. The obtained results reveal that the coding gain advantage of the soft over hard decoding is reduced significantly when throughput is adopted as the performance metric, and it actually vanishes completely for some codes. When adaptive coding is used, the soft decoding advantage is limited to about 1.4 dB.

Error Rate Estimation of Low-Density Parity-Check Codes Decoded by Quantized Soft-Decision Iterative Algorithms

This paper describes a combinatorial approach to estimate the error rate performance of low-density parity-check (LDPC) codes decoded by (quantized) soft-decision iterative decoding algorithms. The method is based on efficient enumeration of input vectors with small distances to a reference vector whose elements are selected to be the most reliable values from the input alphabet. Several techniques, including modified cycle enumeration, and the efficient derivation of problematic inputs for finer quantizers from those of coarser ones are employed to reduce the complexity of the enumeration. The error rate estimate is derived by testing the input vectors of small distances followed by estimating the contribution of larger distance vectors. We demonstrate by a number of examples that the proposed method provides accurate estimates of error rate with computational complexity much lower than that of Monte Carlo simulations, especially at the error floor region.

Iterative Soft-Decision Decoding of Hermitian Codes

This paper proposes an iterative soft-decision decoding algorithm for one of the most popular algebraic-geometric (AG) codes - Hermitian codes. The algorithm is designed by integrating the two most powerful soft-decision decoding algorithms, the adaptive belief propagation (ABP) algorithm and the Koetter-Vardy (KV) list decoding algorithm. The ABP algorithm performs iterative decoding based on an adapted parity-check matrix of a Hermitian code to enhance the reliability of the soft received information. With the enhanced reliability, the KV algorithm performs soft-decision list decoding to obtain the original message. Since the matrix adaptation relies on bit reliabilities, regrouping of the unreliable bits is introduced to assist the ABP decoding. A complexity reducing ABP-KV decoding approach is proposed based on assessing the soft information provided by the ABP algorithm and determining whether the following KV decoding steps should be carried out. Geometric interpretation of the ABP algorithm is presented, demonstrating the necessity of performing matrix adaptation. Our performance analysis shows the proposed iterative decoding algorithm outperforms both the existing decoding approaches for Hermitian codes and the ABP-KV decoding of Reed-Solomon (RS) codes.

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.

Efficient Iterative Techniques for Soft Decision Decoding of Reed-Solomon Codes

Two new iterative soft decision decoding methods for Reed-Solomon (RS) codes are proposed. These methods are based on bit level belief propagation (BP) decoding. In order to make BP decoding effective for RS codes, we use an extended binary parity check matrix with a lower density and reduced number of 4-cycles compared to the original binary parity check matrix of the code. In the first proposed method, we take advantage of the cyclic structure of RS codes. Based on this property, we can apply the belief propagation algorithm on any cyclically shifted version of the received symbols with the same binary parity check matrix. For each shifted version of received symbols, the distribution of reliability values will change and deterministic errors can be avoided. This method results in considerable performance improvement of RS codes compared to hard decision decoding. The performance is also superior to some popular soft decision decoding methods. The second method is based on information correction in BP decoding. It means that we determine least reliable bits and by changing their channel information, the convergence of the decoder is improved. Compared to the first method, this method needs less BP iterations (less complexity) but its performance is not as good.

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.

Iterative Soft-Decision Decoding of Binary Cyclic Codes

Binary cyclic codes achieve good error correction performance and allow the implementation of very simpleencoder and decoder circuits. Among them, BCH codesrepresent a very important class of t-error correcting codes, with known structural properties and error correction capability. Decoding of binary cyclic codes is often accomplished through hard-decision decoders, although it is recognized that softdecision decoding algorithms can produce significant coding gain with respect to hard-decision techniques. Several approaches have been proposed to implement iterative soft-decision decoding of binary cyclic codes. We study the technique based on “extended parity-check matrices”, and show that such method is not suitable for high rates or long codes. We propose a new approach, based on “reduced parity-check matrices” and “spread parity-check matrices”, that can achieve better correction performance in many practical cases, without increasing the complexity.

Construction of Quasi-Cyclic LDPC Codes for AWGN and Binary Erasure Channels: A Finite Field Approach

In the late 1950s and early 1960s, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for hard-decision algebraic decoding, such as Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes. This paper shows that finite fields can also be successfully used to construct algebraic low-density parity-check (LDPC) codes for iterative soft-decision decoding. Methods of construction are presented. LDPC codes constructed by these methods are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN), binary random, and burst erasure channels with iterative decoding in terms of bit-error probability, block-error probability, error-floor, and rate of decoding convergence, collectively. Particularly, they have low error floors. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.

Distributed joint source-channel coding of video using raptor codes

Extending recent works on distributed source coding, this paper considers distributed source-channel coding and targets at the important application of scalable video transmission over wireless networks. The idea is to use a single channel code for both video compression (via Slepian-Wolf coding) and packet loss protection. First, we provide a theoretical code design framework for distributed joint source-channel coding over erasure channels and then apply it to the targeted video application. The resulting video coder is based on a cross-layer design where video compression and protection are performed jointly. We choose Raptor codes - the best approximation to a digital fountain - and address in detail both encoder and decoder designs. Using the received packets together with a correlated video available at the decoder as side information, we devise a new iterative soft-decision decoder for joint Raptor decoding. Simulation results show that, compared to one separate design using Slepian-Wolf compression plus erasure protection and another based on FGS coding plus erasure protection, the proposed joint design provides better video quality at the same number of transmitted packets. Our work represents the first in capitalizing the latest in distributed source coding and near-capacity channel coding for robust video transmission over erasure channels.

On Minimum-WER Performance of a Class of LDPC Codes for the BSC/E Under Extended Bounded-Distance Decoding

We study the word-error-rate (WER) performance and the WER-minimizing decision thresholds plusmnt* for the class of row-column-diagonal (RCD) array codes and their generalizations on the binary symmetric channel with erasures (BSC/E) where decoding is via an extended bounded-distance decoder (e-BDD). This model applies to very-high-data-rate systems where soft-decision iterative decoding is impractical. We demonstrate that substantial WER performance gain can be obtained, especially at higher channel signal-to-noise ratio (CSNR) (CSNR =Delta 10 log10 (1/2sigma2)) values, by employing two optimal thresholds ( plusmnt*) chosen to minimize WER. We study the WER improvement and signal-to-noise ratio (SNR) performance gain as a function of code length (n) for fixed minimum distance (dmin), and as a function of (dmin) for fixed (n). A distinct difference in the behavior of WER versus the threshold t, and the optimal threshold t* versus SNR, is observed between codes with odd and even dmin, with the even dmin. codes exhibiting larger performance improvements when employing a BSC/E instead of a BSC

Near-Capacity Coding in Multicarrier Modulation Systems

We apply irregular low-density parity-check (LDPC) codes to the design of multilevel coded quadrature amplitude modulation (QAM) schemes for application in discrete multitone systems in frequency-selective channels. A combined Gray/Ungerboeck scheme is used to label each QAM constellation. The Gray-labeled bits are protected using an irregular LDPC code with iterative soft-decision decoding, while other bits are protected using a high-rate Reed-Solomon code with hard-decision decoding (or are left uncoded). The rate of the LDPC code is selected by analyzing the capacity of the channel seen by the Gray-labeled bits and is made adaptive by selective concatenation with an inner repetition code. Using a practical bit-loading algorithm, we apply this coding scheme to an ensemble of frequency-selective channels with Gaussian noise. Over a large number of channel realizations, this coding scheme provides an average effective coding gain of more than 7.5 dB at a bit-error rate of 10/sup -7/ and a block length of approximately 10/sup 5/ b. This represents a gap of approximately 2.3 dB from the Shannon limit of the additive white Gaussian noise channel, which could be closed to within 0.8-1.2 dB using constellation shaping.