Efficient Decoding Algorithm Research Articles

Facilitating practical fault-tolerant quantum computing based on color codes

Color codes are a promising topological code for fault-tolerant quantum computing. Insufficient research on color codes has delayed their practical application. In this work, we address several key issues to facilitate practical fault-tolerant quantum computing based on color codes. First, by introducing decoding graphs with error-rate-related weights, we obtain the threshold of 0.47% of the 6.6.6 triangular color code under the standard circuit-level noise model, narrowing the gap to that of the surface code. Second, our work first investigates the circuit-level decoding of color code lattice surgery, then gives an efficient decoding algorithm, which is crucial to perform logical operations in a quantum computer with two-dimensional architectures. Last, a state injection protocol of the triangular color code is proposed, reducing the output magic state error rate in one round of 15 to 1 distillation by two orders of magnitude compared to a previous rough protocol. We also prove that our protocol offers the lowest logical error rates for state injection among all possible codes. Published by the American Physical Society 2024

Implementation of High Speed and Area Efficient Polar Encoder

Abstract: Polar codes were introduced by E.Arıkan in 2008. They are the first family of error-correcting codes that attain the capacity of binary memoryless and symmetric channels with efficient encoding, decoding, and construction algorithms. In this paper, implementation of high speed and area efficient polar encoder for systematic polar codes is presented. According to an iterative property of the generator matrix and particular lower triangular structure of the matrix, the number of XOR computations are reduced. In this implementation, total area of the encoder is decreased by 33.69%, delay is decreased by 54.6% and power consumption is decreased by 39.44%.

Ising model formulation for highly accurate topological color codes decoding

Quantum error correction is an essential ingredient for reliable quantum computation for theoretically provable quantum speedup. Topological color codes, one of the quantum error correction codes, have an advantage against the surface codes in that all Clifford gates can be implemented transversally. However, the hardness of decoding makes the color codes not suitable as the best candidate for experimentally feasible implementation of quantum error correction. Here we propose an Ising model formulation that enables highly accurate decoding of the color codes. In this formulation, we map stabilizer operators to classical spin variables to represent an error satisfying the syndrome. Then we construct an Ising Hamiltonian that counts the number of errors and formulate the decoding problem as an energy minimization problem of an Ising Hamiltonian, which is solved by simulated annealing. In numerical simulations on the (4.8.8) lattice, we find an error threshold of 10.36(5)% for bit-flip noise model, 18.47(5)% for depolarizing noise model, and 2.90(4)% for phenomenological noise model (bit-flip error is located on each of data and measurement qubits), all of which are higher than the thresholds of existing efficient decoding algorithms. Furthermore, we verify that the achieved logical error rates are almost optimal in the sense that they are almost the same as those obtained by exact optimizations by CPLEX with smaller decoding time in many cases. Since the decoding process has been a bottleneck for performance analysis, the proposed decoding method is useful for further exploration of the possibility of the topological color codes. Published by the American Physical Society 2024

Вычисление пар, исправляющих ошибки для алгеброгеометрического кода

For an arbitrary algebrogeometric code and its dual, error-correcting pairs are explicitly calculated. Such a pair consists of codes that are necessary for an efficient decoding algorithm for a given code. The type of pairs depends on the powers of the divisors, with the help of which both the source code and one of the codes included in the pair are constructed. For an algebraic geometric code CL(D, G) of length and associated with a functional field F/Fq of genus g, pairs correcting t = ⌊(n - deg( G) - g - 1)/2⌋ errors, under certain restrictions on the degrees of divisors involved in their construction, are pairs of codes (Cl(D,F),Cl(D,G + F)⊥) or (CL(D,F⊥),Cl(D,F - G)). Restrictions on the degrees of divisors of codes (CL(D,F), CL(D,G - F)) constituting a pair correcting t = ⌊(deg( G) - 3g + 1)/2⌋ errors for the dual code CL(D, G)⊥) Cases where one of the codes involved in constructing a pair belongs to class of MDS codes and the parameters under which this situation is possible are derived. In addition, possible bounds for the divisors involved in the construction of error-correcting pairs for the subfield subcodes CL(D,G)|fp and CL< are calculated /sub>(D, G)⊥)|Fp of the original algebrogeometric code and its dual, with the expansion degree m = 2 (Fq = Fp2).

Twisted Goppa Codes With an Efficient Decoding Algorithm and Quasi-Cyclic Properties

In this paper, we introduce twisted Goppa codes, which generalize classical Goppa codes by adding a twisted term. Then we provide an efficient decoding algorithm for twisted Goppa codes. The Niederreiter cryptosystem is bassed on linear error-correcting codes in which the public key is a parity check matrix. When twisted Goppa codes are applied to the Niederreiter cryptosystem, the public key size is overlarge. To reduce the public key size, we construct quasi-cyclic twisted Goppa codes via a non-trivial automorphism group carefully selecting the defining set and the matched polynomial. Moreover, we obtain a family of cyclic twisted Goppa codes.

Skew differential Goppa codes and their application to Mceliece cryptosystem

A class of linear codes that extends classical Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem.

A Task-Guided Normalized Min-Sum Decoding Network for LDPC Codes-Based DJSCC

Distributed joint source-channel coding (DJSCC) refers to the correlated sources which are encoded separately and reconstructed in the centre node by joint decoding. Therefore, the challenging problem involves designing a simple and efficient decoding algorithm to utilize the correlations among distributed sources. In this letter, a multi-task deep learning decoder is proposed for the low-density parity-check (LDPC) code-based DJSCC system to improve the performance. This proposal is based on the shared neural normalized min-sum (SNNMS) decoding network and the log likelihood ratio (LLR) shared unit is added to further exploit the source correlation. Moreover, an adaptive multi-loss function is proposed to prevent the network tilting the balance in favor of one certain task. Simulation results indicate that the proposed decoder can achieve a significant performance improvement with almost the same complexity compared with the separated SNNMS decoders.

Coding for Polymer-Based Data Storage

Polymer-based data-storage platforms use chains of binary synthetic polymers as recording media and read the content via tandem mass spectrometers. For such systems, we propose the first known family of codes that allows for both unique string reconstruction and correction of multiple mass errors. We consider two approaches: The first approach pertains to asymmetric error-correction and it is based on introducing redundancy that scales linearly with the number of errors and logarithmically with the length of the string. The construction allows for the string to be uniquely reconstructed based only on its erroneous substring composition multiset. The key idea behind our unique reconstruction approach is to interleave (shifted) Catalan-Bertrand strings with arbitrary binary strings and “reflect” them so as to force prefixes and suffixes of the same length to have different weights. The asymptotic code rate of the scheme is one, and decoding is accomplished via a simplified version of the Backtracking algorithm used for the Turnpike problem. For symmetric errors, we use a polynomial characterization of the mass information and adapt polynomial evaluation code constructions for this setting. In the process, we develop new efficient decoding algorithms for a constant number of composition errors.

Basis-Finding Algorithm for Decoding Fountain Codes for DNA-Based Data Storage

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.

Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes

The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm.

Improved Lower Bounds for Strongly Separable Matrices and Related Combinatorial Structures

In nonadaptive group testing, the main research objective is to design an efficient algorithm to identify a set of up to $t$ positive elements among $n$ samples with as few tests as possible. Disjunct matrices and separable matrices are two classical combinatorial structures while one provides a more efficient decoding algorithm and the other needs fewer tests, i.e., larger rate. Recently, a notion of strongly separable matrix has been introduced, which has the same identifying ability as a disjunct matrix, but has larger rate. In this paper, we use a modified probabilistic method to improve the lower bounds for the rate of strongly separable matrices. Using this method, we also improve the lower bounds for some well-known combinatorial structures, including locally thin set families and cancellative set families.

A Construction of Maximum Distance Profile Convolutional Codes With Small Alphabet Sizes

Convolutional codes are essential in a wide range of practical applications due to their efficient non-algebraic decoding algorithms. In this paper, we first propose a new family of matrices over finite fields by combining Vandermonde and Moore matrices. Using favourable properties of the matrices in this new family enables us to construct a new family of convolutional codes with memory 1 and maximum distance profile. It is notable that the alphabet sizes of this new family of convolutional codes with maximum distance profile can be kept significantly smaller than those in the literature. Keeping the code rate to a constant, the alphabet size is roughly the square root of the previously best-known value.

The syndromes decoding algorithm in group codes

Our aim is the design of an efficient decoding algorithm in group codes. The algorithm is inspired by the well known syndrome decoding algorithm for linear codes and uses the decomposition of a semisimple group algebra KG as a direct sum of two-sided ideals, each of them generated by a central idempotent of KG. When G is an abelian group the algorithm can be modified to make it very simple and efficient. Some illustrative examples are presented.

Efficient Decoding Algorithm for Binary Quadratic Residue Codes Using Reduced Permutation Sets

The Quadratic Residue (QR) codes have a rich mathematic structure. Unfortunately, their Algebraic Decoding (AD) is not generalizable for all QR codes. In this study, an efficient hard decoding algorithm is proposed to generalize the decoding of the binary systematic Quadratic Residue (QR) codes. The proposed decoder corrects t erroneous bits or less, in the received word, based on a reduced set of permutations derived from the large automorphism group of QR codes. This set of permutations is applied to the received word to move the error positions and trap all of them in redundancy. Then, to evaluate the proposed method, we applied it to many binary QR codes of moderate code length starting with 17 until 113 with reducible and irreducible generator polynomials. The proposed decoder was validated by inserting all possible error patterns, that have t or less erroneous positions, as input of the proposed decoder and the output is always a correct codeword. The complexity study, in terms of the number of operations used, reveals that the light permutation decoding LPD algorithm significantly decreases decoding complexity without performance loss. So, it is qualified to be a good competitor to decode QR codes with lower lengths but is the best for QR codes with higher lengths.

A Flexible Encoding/Decoding Procedure for 6G SCMA Wireless Networks via Adversarial Machine Learning Techniques

5 G networks may not be competent in fully supporting the high data rates, system capacity, and spectral efficiency expected in the upcoming 6 G networks. An enabling factor to achieve these requirements is the adoption of the Sparse Code Multiple Access (SCMA), which is one of the most promising code-domain Non-Orthogonal Multiple Access (NOMA) techniques. Nowadays, there are two major challenges for the implementation of SCMA in practical networks. First, the design of a near-optimal codebook set at the transmitter, that heavily impacts the network performance. Second, an efficient decoding algorithm at the receiver, that dominates the computational complexity of the system. The two aspects are not independent from each other; therefore the optimization of an SCMA network should address jointly both the Encoder and the Decoder side. Moreover, such solution is far from optimal when designed for general purpose use, since it does not exploit the channel statistical information known at the transmitter and the receiver side. However, most of the works available in the literature address only one aspect and also are not tailored for a given radio channel. In this paper, we design an end-to-end SCMA en/deconding structure based on the integration between a state-of-the-art AutoEncoder (AE) architecture and a novel Wasserstein Generative Adversarial Network (WGAN). The aim is to jointly create a near-optimal codebook design and a non-computationally heavy Decoder that are robust to the given channel statistics. Finally, we trained our model to be flexible to support different channel conditions and compared its performance against the state-of-the-art AE and the best conventional Decoders. The simulation results show that our innovative design exhibits decoding performance very similar to conventional Decoders with a reduced decoding latency.

Efficient MP Decoding via Fast G-BADMM Approach for Binary LDPC Codes

Mathematical programming (MP) decoding based on the alternate direction method of multipliers (ADMM) algorithm is a promising approach for low-density parity-check (LDPC) codes in recent years. For this kind of decoding, the convergence characteristic has been considered as a key factor to enhance its advantage in practice. In this letter, we propose an efficient decoding algorithm based on generalized Bregman ADMM (G-BADMM) technique to improve the convergence speed of ADMM-based MP decoding for binary LDPC codes. The G-BADMM algorithm is established by adding additional Bregman divergences to the ADMM iterations of MP decoding. Moreover, the resulted G-BADMM iterations are solved efficiently. Furthermore, we theoretically analyze the proposed G-BADMM algorithm including its decoding performance analysis and computational complexity. Simulations demonstrate the efficiency of the proposed G-BADMM-based decoding algorithm.

Recent Advances in Fatigue Detection Algorithm Based on EEG

Fatigue is a state commonly caused by overworked, which seriously affects daily work and life. How to detect mental fatigue has always been a hot spot for researchers to explore. Electroencephalogram (EEG) is considered one of the most accurate and objective indicators. This article investigated the development of classification algorithms applied in EEG-based fatigue detection in recent years. According to the different source of the data, we can divide these classification algorithms into two categories, intra-subject (within the same subject) and cross-subject (across different subjects). In most studies, traditional machine learning algorithms with artificial feature extraction methods were commonly used for fatigue detection as intra-subject algorithms. Besides, deep learning algorithms have been applied to fatigue detection and could achieve effective result based on large-scale dataset. However, it is difficult to perform long-term calibration training on the subjects in practical applications. With the lack of large samples, transfer learning algorithms as a cross-subject algorithm could promote the practical application of fatigue detection methods. We found that the research based on deep learning and transfer learning has gradually increased in recent years. But as a field with increasing requirements, researchers still need to continue to explore efficient decoding algorithms, design effective experimental paradigms, and collect and accumulate valid standard data, to achieve fast and accurate fatigue detection methods or systems to further widely apply.

Spatially Coupled PLDPC-Hadamard Convolutional Codes

We propose a new type of ultimate-Shannon-limit-approaching codes called spatially coupled protograph-based low-density parity-check Hadamard convolutional codes (SC-PLDPCH-CCs), which are constructed by spatially coupling PLDPC-Hadamard block codes. We develop an efficient decoding algorithm that combines pipeline decoding and layered scheduling for the decoding of SC-PLDPCH-CCs, and analyze the latency and complexity of the decoder. To estimate the decoding thresholds of SC-PLDPCH-CCs, we first propose a layered protograph extrinsic information transfer (PEXIT) algorithm to evaluate the thresholds of spatially coupled PLDPC-Hadamard terminated codes (SC-PLDPCH-TDCs) with a moderate coupling length. With the use of the proposed layered PEXIT method, we develop a genetic algorithm to find good SC-PLDPCH-TDCs in a systematic way. Then we extend the coupling length of these SC-PLDPCH-TDCs to form good SC-PLDPCH-CCs. Results show that our constructed SC-PLDPCH-CCs can achieve comparable thresholds to the block code counterparts. Simulations illustrate the superiority of the SC-PLDPCH-CCs over the block code counterparts and other state-of-the-art low-rate codes in terms of error performance. For the rate-0.00295 SC-PLDPCH-CC, a bit error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−5</sup> is achieved at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$E_{b}/N_{0} = -1.465$ </tex-math></inline-formula> dB, which is only 0.125 dB from the ultimate Shannon limit.

Morphing Quantum Codes

We introduce a morphing procedure that can be used to generate new quantum codes from existing quantum codes. In particular, we morph the 15-qubit Reed-Muller code to obtain a $[\![10,1,2]\!]$ code that is the smallest known stabilizer code with a fault-tolerant logical $T$ gate. In addition, we construct a family of hybrid color-toric codes by morphing the color code. Our code family inherits the fault-tolerant gates of the original color code, implemented via constant-depth local unitaries. As a special case of this construction, we obtain toric codes with fault-tolerant multi-qubit control-$Z$ gates. We also provide an efficient decoding algorithm for hybrid color-toric codes in two dimensions, and numerically benchmark its performance for phase-flip noise. We expect that morphing may also be a useful technique for modifying other code families such as triorthogonal codes.

Design a Novel BCI for Neurorehabilitation Using Concurrent LFP and EEG Features: A Case Study.

Brain-computer interfaces (BCI) that enables people with severe motor disabilities to use their brain signals for direct control of objects have attracted increased interest in rehabilitation. To date, no study has investigated feasibility of the BCI framework incorporating both intracortical and scalp signals. Concurrent local field potential (LFP) from the hand-knob area and scalp EEG were recorded in a paraplegic patient undergoing a spike-based close-loop neurorehabilitation training. Based upon multimodal spatio-spectral feature extraction and Naïve Bayes classification, we developed, for the first time, a novel LFP-EEG-BCI for motor intention decoding. A transfer learning (TL) approach was employed to further improve the feasibility. The performance of the proposed LFP-EEG-BCI for four-class upper-limb motor intention decoding was assessed. Using a decision fusion strategy, we showed that the LFP-EEG-BCI significantly (p 0.05) outperformed single modal BCI (LFP-BCI and EEG-BCI) in terms of decoding accuracy with the best performance achieved using regularized common spatial pattern features. Interrogation of feature characteristics revealed discriminative spatial and spectral patterns, which may lead to new insights for better understanding of brain dynamics during different motor imagery tasks and promote development of efficient decoding algorithms. Moreover, we showed that similar classification performance could be obtained with few training trials, therefore highlighting the efficacy of TL. The present findings demonstrated the superiority of the novel LFP-EEG-BCI in motor intention decoding. This work introduced a novel LFP-EEG-BCI that may lead to new directions for developing practical neurorehabilitation systems with high detection accuracy and multi-paradigm feasibility in clinical applications.