A Low-Complexity Ordered Statistic Decoding of Short Block Codes

Polar codes have been used to guarantee high reliability in communication due to their excellent performance. In this letter, we propose a guessing error pattern ordered statistic decoding (GEPOSD) algorithm to further improve the performance of short polar codes. It outperforms the widely used cyclic redundancy check-aided successive cancellation list (CA-SCL) decoding under a code length of 64. The GEPOSD algorithm utilizes a strategy based on the maximum a posteriori probability rule to select the error pattern on the most reliable independent bits. Subsequently, an error pattern set is extended using a tree search approach. To minimize computational complexity, we introduce two early stopping criteria that limit the number of error patterns tested in decoding. The simulation results show that the complexity of GEPOSD is significantly reduced compared to the original OSD algorithm.

In this contribution, KNIH algorithm and Ordered Statistic Decoding (OSD) algorithm are investigated, based on which, a novel KNIH-OSD parallel decoding scheme for LDPC codes is proposed. KNIH decoding algorithm processes certain LRPs of a received sequence, while OSD algorithm processes certain MRIPs of a received sequence. If there are too many errors in MRIPs, the OSD algorithm will fail, on the contrary, if there are too many errors in LRPs, the KNIH algorithm will fail. This contribution proposes a parallel decoding scheme based on the complementary characteristic of these two algorithms. Simulation results show that the proposed scheme is feasible and effective. Compared with OSD and KNIH algorithm, the decoding performance is improved.

The conventional soft decision decoding (SDD) methods require various hard decision decoders (HDDs) based on different codes or re-manipulate the generator matrix by the complicated Gaussian elimination technique according to the bit reliability. This paper presents a general multi-class neural network (NN)-based decoder for the short linear block codes, where no HDD and Gaussian elimination are required once the NN is constructed. This network architecture performs multi-classification to select the messages with high occurrence probabilities and chooses the best codeword on a maximum likelihood basis. Simulation results show that the developed approach outperforms the existing deep neural network (DNN)-based decoders in terms of decoding time and bit error rate (BER). The error-correcting performance is also superior to the conventional Chase-II algorithm and is close to the ordered statistics decoding (OSD) in most cases. For Bose–Chaudhuri–Hocquenghem (BCH) codes, the SNR is improved by 1dB to 4dB as the BER is 10−4. For the (23, 12) quadratic residue (QR) code, the SNR is improved by 2dB when the BER is 10−3. The developed NN-based decoder is quite general and applicable to various short linear block codes with good BER performance.

In this work, we present an improvement on low complexity serial concatenation of belief propagation (BP) - order statistic decoding (OSD) algorithm for low-density parity-check (LDPC) codes. We introduce a more reliable method to reconstruct ordered information sequence in terms of the accumulated log-likelihood ratio (LLR) transitions of variable nodes. We give a general expression for both the existing BP-OSD algorithm and our proposed new method according to a structure similar to an infinite impulse response (IIR) filter. This improved algorithm achieves noticeable performance gains with only modest increase in computation complexity.

Guessing Random Additive Noise Decoding (GRAND) excels at decoding high-rate codes but struggles to decode low-rate codes with reasonable complexity. Ordered Statistics Decoding (OSD) specifically excels in decoding short codes irrespective of rates; however, OSD necessitates the use of Gaussian elimination which introduces additional time, space and computational complexity. Partial Ordered Statistics Decoding (POSD) was proposed to reduce the time, space, and computational complexity of OSD; however, the current partition-based POSD has poor decoding performance since it does not generate test error patterns across partitions. In this paper, we propose to improve the decoding performance of POSD by incorporating test error patterns inspired by GRAND methods. This work offers a trade-off between performance and complexity compared to existing decoders such as GRAND and OSD. We enhance POSD by optimizing the scheduling of Test Error Patterns (TEPs) and show that our technique can be applied to any code in a standard form. At a target BER 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−4</sup> with eBCH (128,64) the enhanced error patterns achieve more than 0.6 dB gain in performance compared to the POSD with partition-based error patterns. Moreover, at a target frame error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−5</sup> , POSD uses 10× less binary operations compared to GRAND when decoding eBCH (128,64) and RLC(128,64) codes. With BCH (127,29) and RLC(128,32), at a target frame error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−2</sup> , POSD with enhanced error patterns with a maximum number of queries (MQ) of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> achieves up to a 2 dB gain to its GRAND equivalent which is using 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sup> maximum number of queries.

In this paper, a novel stopping criterion for Belief Propagation (BP) decoding of polar codes is proposed. The proposed stopping criterion is based on the Bit Difference Ratio (BDR). The BDR stopping method is implemented by monitoring the fluctuation of BDR along with the number of iterations, to decide if the BP decoding process can be terminated. Simulation results indicate that the proposed criterion can provide superiority in iteration reduction compared with the current stopping criteria. Hence, the proposed BDR stopping criterion can effectively reduce the energy waste and decoding latency in systems. Moreover, by loosing the constraint, the proposed stopping criterion can achieve more iteration reduction at the cost of Bit Error Rate (BER) degradation.

Irregular LDPC codes can achieve better error rate performance than regular LDPC codes. However, irregular LDPC codes have higher error floors than regular LDPC codes. The Ordered Statistic Decoding (OSD) algorithm achieves approximate Maximum Likelihood (ML) decoding. ML decoding is effective to lower error floors. However, the OSD estimates satisfy the parity check equation of the LDPC code even the estimates are wrong. Hybrid decoder combining LLR-BP decoding algorithm and the OSD algorithm cannot also lower error floors, because wrong estimates also satisfy the LDPC parity check equation. We proposed the concatenated code constructed with an inner irregular LDPC code and an outer Cyclic Redundancy Check (CRC). Owing to CRC, we can detect wrong codewords from OSD estimates. Our CRC-LDPC code with hybrid decoder can lower error floors in an AWGN channel. In wireless communications, we cannot neglect the effects of the channel. The OSD algorithm needs the ordering of each bit based on the reliability. The Channel State Information (CSI) is used for deciding reliability of each bit. In this paper, we evaluate the Block Error Rate (BLER) of the CRC-LDPC code with hybrid decoder in a fast fading channel with perfect and imperfect CSIs where 'imperfect CSI' means that the distribution of channel and those statistical average of the fading amplitudes are known at the receiver. By computer simulation, we show that the CRC-LDPC code with hybrid decoder can lower error floors than the conventional LDPC code with hybrid decoder in the fast fading channel with perfect and imperfect CSIs. We also show that combining error detection with the OSD algorithm is effective not only for lowering the error floor but also for reducing computational complexity of the OSD algorithm.

Multi-objective genetic algorithms have been successfully applied in a wide variety of problems. Although widely used, there are few theoretical guidelines for determining when to stop the search. Many users commonly use rules like stopping when there is no significant improvement during the last generations or when a certain number of generations are reached. In this paper, we propose a new stopping criterion approach and evaluate its performance with three widely used evolutionary algorithms in the calibration of a hydrologic model. The stopping criterion is based on the minimum number of generations required to achieve a determined number of non-dominated solutions in Pareto Front. The new stopping criterion was tested in the lumped hydrologic model IPH-II calibration, using the genetic algorithms NSGA-II, NSGA-III, and SPEA-II and two objective functions. The generational distance, spacing, and maximum spread metrics were used to assess the performance of the proposed stopping criterion in comparison to the standard criterion. Results show no significant loss in goodness of fit associated with the proposed stopping criterion, both in calibration and validation periods. Performance metrics have shown similar values when the standard and the proposed stopping criteria were compared. However, the average computational time to complete the optimization process was reduced up to 38.2% when the proposed stopping criterion was used. Thus, it can be concluded that the new stopping criterion reduces the iteration workload without compromising the accuracy of solution sets.

In this paper, we show how the presence of locality within a binary cyclic code can be exploited to improve decoding performance and to reduce decoding complexity. We pursue two approaches. Under the first approach, we show how the ordered statistics decoding (OSD) method can be modified by inserting a simple single round belief-propagation step at the start that involves only the local codes. The resultant locality-aware OSD algorithm yields an appreciable signal-to-noise ratio (SNR) gain for a given level of reliability and essentially the same level of decoder complexity. Under the second, trellis decoding approach, we show that the careful introduction of locality results in the creation of a cyclic subcode that possesses lower maximum state complexity. In addition, we present a simple means of deriving an upper bound to the state complexity profile of any cyclic code that is based only on the zeros of the code. Furthermore, we show how the decoding speed of either locality-aware OSD or trellis decoding can be significantly increased in the presence of locality, in the moderate-to-high SNR regime, by making the use of a quick-look decoder that often returns the maximum likelihood code word.

Data center networks (DCNs) require a low-cost, low-power optical transceiver to handle increased traffic from generative artificial intelligence, video streaming services, and more. Improving the required signal-to-noise ratio (RSNR) by digital signal processing such as forward error correction (FEC) mitigates the requirements for electrical and optical components. The optical transceivers in DCNs exploit a low-complexity soft-decision (SD) FEC, consisting of short block-length linear error-correcting codes and a low-complexity SD decoder (SDD), such as Chase decoding and ordered statistics decoding. The low-complexity SDD efficiently approaches a maximum likelihood decoding (MLD). However, the decoding performance of MLD is limited by its finite block length. In this paper, we describe the details of our proposed channel-polarized multilevel coding with iterative decoding (CP-MLC-ID). The proposed CP-MLC-ID improves the decoding performance by extending the codeword length to weakly and indirectly connect codewords via bypassed bits. The 19.5% –OH CP-MLC-ID using 128-bit extended Bose–Chaudhuri–Hocquenghem (eBCH) and KP4 codes outperforms the concatenated eBCH and KP4 codes with a net coding gain of 0.25 and 0.40dB for the same and double the number of SDDs, respectively. We also investigate the dependency of the decoding performance on the size of a bit interleaver. The performance degradation of CP-MLC-ID using an 8-bit interleaver is about 0.1dB compared to using the large-bit interleaver. Our results indicate that even a weak connection by exclusive-OR between codewords improves the decoding performance, compared to simple concatenated codes in the DCNs.

In this paper, we consider a broadcast application for mobile terminals where the access points are connected to the source via a fixed network. Random Linear Network (RLN) coding is used in the fixed network to maximize its transmission rate and a wireless connection is used between the access points and the terminal. In order to reduce variations in the transmission delay and offer more flexibility to the terminal, we perform the network decoding in the terminal. We investigate the error correction capability of RLN coding using the ordered statistics decoding (OSD) algorithm. Taking into account the forward error correcting (FEC) code used on the wireless link, a product code is built by combining the FEC code and the RLN code and turbo decoded in the terminal. The proposed scheme efficiency is analyzed through its simulated and theoretical performance under additive white Gaussian noise (AWGN) and non frequency-selective block Rayleigh fading channels.

The latest research progress of the theory of compressed sensing (CS) over graphs makes it possible that the advantage of CS can be utilized by data ferries to gather data in WSNs. In this paper, we leverage the non-uniform distribution of the sensing data field to significantly reduce the required number of data ferries, yet ensuring the recovered data quality. Specially, we propose an intelligent compressive data gathering scheme consisting of an efficient stopping criterion and a novel learning strategy. The proposed stopping criterion is based only on the gathered data, without relying on the priori knowledge on the sparsity of unknown sensing data. Our strategy minimizes the number of data ferries while guaranteeing the data quality by learning the statistical distribution of gathered data. Simulation results show that the proposed scheme improves the reconstruction quality compared to the existing ones.

Blind image deblurring algorithms (BIDA) constitute a subset of image restoration algorithms used to solve ill-posed inverse problems, which can be challenging in low SNR situations. Those iterative methods usually impose some regularization upon the deconvolution process in order to constrain the problem and reduce the size of solution space. In this paper, we propose to use a new stopping criterion for iterative deblurring algorithms based on our previously published no reference blur image quality measure. The rationale behind the proposed stopping criterion is to control the deconvolution process by estimating the reminding blur quantity. For test, two iterative deblurring algorithms are considered. The Lucy Richardson [1,2] and the Shock Filters methods [3,4]. The proposed adaptive approach has been tested on blurred images from LIVE database (Gblur).

The latest research progress of the theory of compressive sensing (CS) over graphs makes it possible that the advantage of CS can be utilized by data ferries to gather data for wireless sensor networks. In this paper, we leverage the non-uniform distribution of the sensing data field to significantly reduce the required number of data ferries, yet ensuring the recovered data quality. Specially, we propose an intelligent compressive data gathering scheme consisting of an efficient stopping criterion and a novel learning strategy. The proposed stopping criterion is based only on the gathered data, without relying on the priori knowledge on the sparsity of unknown sensing data. Our learning strategy minimizes the number of data ferries while guaranteeing the data quality by learning the statistical distribution of the gathered data. Simulation results show that the proposed scheme improves the reconstruction accuracy and stability compared to the existing ones.

This paper presents a systematic study of early stopping criteria for Turbo decoding. First, statistical analysis is carried out on numerous hard/ soft variables that may be used in an early stopping criterion. Desirable variables are suggested based on their statistical properties. Simulation results show that any stopping criteria based on a single variable will have BER/FER performance loss. Two criteria, each of which uses two variables, are recommended in this paper and neither of them will result in any performance loss. It is also shown in this paper that the thresholds for these variables should be set to be proportional to the logarithm of the block size instead of being proportional to the block size.