Introducing Ambiguity Clustering: An Accurate and Efficient Decoder for qLDPC Codes

The problem of efficient maximum-likelihood soft decision decoding of binary BCH codes is considered. It is known that those primitive BCH codes whose designed distance is one less than a power of two, contain subcodes of high dimension which consist of a direct-sum of several identical codes. The authors show that the same kind of direct-sum structure exists in all the primitive BCH codes, as well as in the BCH codes of composite block length. They also introduce a related structure termed the concurring-sum, and then establish its existence in the primitive binary BCH codes. Both structures are employed to upper bound the number of states in the minimal trellis of BCH codes, and develop efficient algorithms for maximum-likelihood soft decision decoding of these codes. >

By introducing the ideas of reducing computation complexity in the fast Fourier transform (FFT) to decoding, a low complexity decoding of binary BCH codes is derived which transforms the determination of one-dimensional error word into two-dimensional matrix. For the case of double-error-correcting binary BCH codes, the frequency domain decoding based on prime-factor FFT (in short, PF-FDD) was designed. Compared with the conventional frequency domain decoding algorithm, PF-FDD has the advantages of lower decoding complexity and faster decoding speed at the expense of increasing storage memory. When compared with the ordinary look-up table decoding scheme, the error patterns to store are decreased greatly. Hence, PF-FDD achieves a better tradeoff between decoding complexity and hardware complexity. The proposed low complexity decoding scheme is suitable for binary BCH codes whose block length has prime factors and error correcting capacity is relatively small, and is expected to apply to soft-decision decoding.

This paper investigates high-performance decoder design for regular and irregular repeat-accumulate (RA) codes of large block length. In order to achieve throughputs and bit-error rate performance that are inline with future trends in high-speed communications. high-throughput and low-power decoders of low complexity are needed. To meet such conflicting requirements for long codes, the concept of architecture-aware RA (AARA) code design is proposed. AARA code design decouples the complexity of the decoder from the owe structure by inducing structural regularity features that are amenable to efficient and scalable decoder implementations. Design methods of AARA codes with structured permuters for which an iterative decoding algorithm performs well under message-passing are analogous to those for AA LDPC codes. Algorithmic and architectural optimizations that address the latency, memory overhead, and complexity problems typical of iterative decoders for long RA codes are investigated, and a staggered decoding schedule is introduced. AARA decoders using the proposed schedule have substantial advantage over serial and parallel RA decoders.

Decoding motor imagery (MI) from electroencephalogram (EEG) signals is a cornerstone of brain-computer interface (BCI) systems. However, existing methods often face a critical trade-off between decoding accuracy and model interpretability, limiting their applicability in real-world settings. To address this challenge, we proposed the WavTSK, an end-to-end interpretable fuzzy neural network for efficient MI-EEG decoding. The WavTSK employs a deeply integrated architecture that combines a learnable wavelet-based feature extraction module with a multi-rule Takagi-Sugeno-Kang (TSK) fuzzy classifier. The feature extractor incorporates learnable wavelet filters, statistical descriptors, and an adaptive band-weighting mechanism to capture rich multi-scale time-frequency representations directly from raw EEG. The extracted representations are then processed by a TSK fuzzy reasoning layer, enabling rule-level transparency in decoding. Experiments on the four-class BCI Competition IV-2a dataset showed that WavTSK achieved mean decoding accuracies of 71.63% in cross-validation and 70.55% in the cross-session hold-out setting, consistently outperforming state-of-the-art black-box deep learning models while maintaining strong interpretability. The results highlight the potential of WavTSK as a powerful and interpretable framework for reliable EEG decoding, advancing the development of trustworthy and clinically applicable BCI technologies.

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.

Color codes are a class of topological quantum codes with a high error threshold and a large set of transversal encoded gates and are thus suitable for fault-tolerant quantum computation in two-dimensional architectures. Recently, computationally efficient decoders for the color codes were proposed. We describe an alternate efficient iterative decoder for topological color codes and apply it to the color code on the hexagonal lattice embedded on a torus. In numerical simulations, we find an error threshold of 7.8$%$ for independent dephasing and spin-flip errors.

A systematic design of a trellis-based maximum-likelihood soft-decision decoder for linear block codes is presented. The essence of the decoder is to apply an efficient search algorithm for the error pattern on a reduced trellis representation of a certain coset. Rather than other efficient decoding algorithms, the proposed decoder is systematically designed for long codes, as well as for short codes. Computational gain of up to 6 is achieved for long high-rate codes over the well-known trellis decoder of Wolf (1978). Efficient decoders are also obtained for short and moderate length codes. >

Measures of decoding and oral language have been shown to predict early reading comprehension across a wide variety of languages, though the timeframe and strength of the predictions vary by orthographic depth. This study is the first to examine predictors of early reading in Vietnamese, a transparent orthography of Romanized letters and diacritics. Eighty-two children in Hanoi, Vietnam, completed measures of decoding and oral language in kindergarten (phonological awareness, PA; rapid automatized naming, RAN; receptive and expressive vocabulary) and measures of decoding and reading comprehension in first grade. Average performance at the end of first grade, after just one year of formal instruction, was near ceiling on word reading but more variable on nonword and text reading. Kindergarten PA and RAN (but not vocabulary) predicted first-grade decoding after controlling for maternal education and kindergarten decoding, and PA was a stronger predictor than RAN (10% vs 4% of unique variance). The best predictors of first grade reading comprehension were first-grade decoding (47% of unique variance) and kindergarten expressive vocabulary (4% of unique variance) after controlling for maternal education. Overall, Vietnamese children became accurate and efficient decoders after one year of instruction. Findings from RAN and PA suggest their utility in guiding differentiated instruction on decoding. Kindergarten vocabulary, which differed as a function of maternal education, significantly predicted first-grade comprehension.

In this paper, we consider the decoding of fountain codes where the received symbols may have errors. It is motivated by the application of fountain codes in DNA-based data storage systems where the inner code decoding, which generally has undetectable errors, is performed before the outer fountain code decoding. We propose a novel and efficient decoding algorithm, namely basis-finding algorithm (BFA), followed by three implementations. The key idea of the BFA is to find a basis of the received symbols, and then use the most reliable basis elements to recover the source symbols with the inactivation decoding. Gaussian elimination is used to find the basis and to identify the most reliable basis elements. As a result, the BFA has polynomial time complexity. For random fountain codes, we are able to derive some theoretical bounds for the frame error rate (FER) of the BFA. Extensive simulations with Luby transform (LT) codes show that, the BFA has significantly lower FER than the belief propagation (BP) algorithm except for an extremely large amount of received symbols, and the FER of the BFA generally decreases as the average weight of basis elements increases.

Efficient list decoding of Reed-Solomon codes beyond their error-correction radius has been studied in a number of recent papers. The approach is based on algebraic weighted-interpolation techniques. In this paper, we develop weight assignment schemes for arbitrary additive cost functions. Such functions are defined on the product space F/sub q//spl times//spl Yscr/ They include the Hamming metric and the generalized Hamming metric, as well as log-likelihood based costs, as special cases.

When the code is not long, non-binary low-density parity-check (NB-LDPC) codes can achieve better error-correcting performance than binary LDPC codes at the cost of higher decoding complexity. The recently developed iterative reliability-based majority-logic NB-LDPC decoding can achieve better performance-complexity tradeoffs than previous algorithms. Many existing NB-LDPC code construction schemes lead to quasi-cyclic or cyclic codes. In this paper, efficient low-complexity NB-LDPC decoder architectures are developed for these two types of codes based on the newly proposed iterative hard reliability-based majority-logic decoding (IHRB-MLGD). Particularly, novel schemes are designed to keep a small proportion of messages in order to reduce the memory requirement without causing noticeable performance loss. Moreover, a shift-message structure is proposed by using memories concatenated with variable node units to enable efficient partial-parallel decoding for cyclic NB-LDPC codes. Compared to previous decoders based on the Min-max algorithm, the proposed IHRB-MLGD decoder architectures can achieve tens of times higher efficiency for codes with similar length and rate with moderate coding gain loss.

It is shown that the relatively unknown Dorsch decoder may be extended to produce a decoder that is capable of maximum-likelihood decoding. The extension involves a technique for any linear (n, k) code that ensures that n−k less reliable, soft decisions of each received vector may be treated as erasures in determining candidate codewords. These codewords are derived from low information weight codewords and it is shown that an upper bound of this information weight may be calculated from each received vector in order to guarantee that the decoder will achieve maximum-likelihood decoding. Using the cross-correlation function, it is shown that the most likely codeword may be derived from a partial correlation function of these low information weight codewords, which leads to an efficient fast decoder. For a practical implementation, this decoder may be further simplified into a concatenation of a hard-decision decoder and a partial correlation decoder with insignificant performance degradation. Results are presented for some powerful, known codes, including a GF(4) non-binary BCH code. It is shown that maximum-likelihood decoding is realised for a high percentage of decoded codewords and that performance close to the sphere packing bound is attainable for codeword lengths up to 1000 bits.

Low-density parity-check (LDPC) code as a very promising error-correction code has been adopted as the channel coding scheme in the fifth-generation (5G) new radio. However, it is very challenging to design a high-performance decoder for 5G LDPC codes because their inherent numerous degree-1 variable-nodes are very prone to be erroneous. In this article, the problem is solved gracefully by developing a low-complexity check-node update function, greatly improving the reliability of check-to-variable messages. By further incorporating the proposed column degree adaptation strategy, our decoder could offer a 0.4dB performance gain over the existing ones. In addition, this article presents an efficient 5G LDPC decoder architecture. Benefiting the specific structure of 5G LDPC codes, layer merging, split storage method, and selective-shift structure are introduced to facilitate a significant reduction of decoding delay and area consumption. Implementation result on 90-nm CMOS technology demonstrates that the proposed decoder architecture yields an impressive improvement in throughput-to-area ratio, achieving up to 173.3% compared to conventional design.

In this paper, we propose efficient maximum-likelihood (ML) decoding for binary Kronecker product-based (KPB) codes. This class of codes, have a matrix defined by the n-fold iterated Kronecker product Gn = F⊗n of a binary upper-triangular kernel matrix F, where some columns are suppressed given a specific puncturing pattern. Polar and Reed- Muller codes are well known examples of such KPB codes. The triangular structure of Gn enables to perform ML decoding as a binary tree search for the closest codeword to the received point. We take advantage of the highly regular fractal structure of Gn and the “tree folding” technique to design an efficient ML decoder, enabling to decode relatively longer block lengths than with a standard binary tree search. The tree κ-folding operation transforms the binary tree with N levels into a non-binary tree with N=2κ levels, where the search can be significantly accelerated by a suitable ordering of the branch metrics. For a given κ we can find (n over κ) different folding which lead to decoders with different complexity, for a given code. Using the proposed folded tree decoder, we provide exact ML performances of some Reed-Muller and polar codes over a binary AWGN channel for the block length up to 256.

In Chapter 6, we introduced an efficient decoder for GRS codes, yet we assumed that the number of errors does not exceed ⌊( d −1)/2⌋, where d is the minimum distance of the code. In this chapter, we present a decoding algorithm for GRS codes, due to Guruswami and Sudan, where this upper limit is relaxed. When a decoder attempts to correct more than ⌊( d −1)/2⌋ errors, the decoding may sometimes not be unique; therefore, we consider here a more general model of decoding, allowing the decoder to return a list of codewords, rather than just one codeword. In this more general setting, a decoding is considered successful if the computed list of codewords contains the transmitted codeword. The (maximum) number of errors that a list decoder can successfully handle is called the decoding radius of the decoder. The approach that leads to the Guruswami–Sudan list decoder is quite different from the GRS decoder which was introduced in Chapter 6. Specifically, the first decoding step now computes from the received word a certain bivariate polynomial Q(x, z) over the ground field, F , of the code. Regarding Q(x, z) as a univariate polynomial in the indeterminate z over the ring F[x] , a second decoding step computes the roots of Q(x, z) in F[x] ; these roots are then mapped to codewords which, in turn, form the returned list.