In this paper, we consider quantum error correction over depolarizing channels with non-binary low-density parity-check codes defined over Galois field of size $2^p$ . The proposed quantum error correcting codes are based on the binary quasi-cyclic CSS (Calderbank, Shor and Steane) codes. The resulting quantum codes outperform the best known quantum codes and surpass the performance limit of the bounded distance decoder. By increasing the size of the underlying Galois field, i.e., $2^p$, the error floors are considerably improved.
We propose a modified iterative bounded distance decoding of product codes. The proposed algorithm is based on exchanging hard messages iteratively and exploiting channel reliabilities to make hard decisions at each iteration. Performance improvements up to 0.26 dB are achieved.
We use density evolution to optimize the parameters of binary product codes (PCs) decoded based on the recently introduced iterative bounded distance decoding with scaled reliability. We show that binary PCs with component codes of 3-bit error correcting capability provide the best performance-complexity trade-off.
Raptor codes are known as precoded rateless codes that achieve the capacity of BEC. However the maximum degree of Raptor codes needs to be unbounded to achieve the capacity. In this paper, we prove that spatially-coupled precoded rateless codes achieve the capacity with bounded degree under BP decoding.
Affine Subcode Ensemble Decoding for Degeneracy-Aware Quantum Error Correction
Communications Engineering Lab (CEL), Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院通信工程实验室）；School of Informatics, University of Edinburgh（爱丁堡大学信息学院）
Decoding error-correctiong codes by methods of mathematical optimization, most importantly linear programming, has become an important alternative approach to both algebraic and iterative decoding methods since its introduction by Feldman et al. At first celebrated mainly for its analytical powers, real-world applications of LP decoding are now within reach thanks to most recent research. This document gives an elaborate introduction into both mathematical optimization and coding theory as well as a review of the contributions by which these two areas have found common ground.
We investigate threshold-based multi-trial decoding of concatenated codes with an inner Maximum-Likelihood decoder and an outer error/erasure (L+1)/L-extended Bounded Distance decoder, i.e. a decoder which corrects e errors and t erasures if e(L+1)/L + t <= d - 1, where d is the minimum distance of the outer code and L is a positive integer. This is a generalization of Forney's GMD decoding, which was considered only for L = 1, i.e. outer Bounded Minimum Distance decoding. One important example for (L+1)/L-extended Bounded Distance decoders is decoding of L-Interleaved Reed-Solomon codes. Our …
A Viterbi-like decoding algorithm is proposed in this paper for generalized convolutional network error correction coding. Different from classical Viterbi algorithm, our decoding algorithm is based on minimum error weight rather than the shortest Hamming distance between received and sent sequences. Network errors may disperse or neutralize due to network transmission and convolutional network coding. Therefore, classical decoding algorithm cannot be employed any more. Source decoding was proposed by multiplying the inverse of network transmission matrix, where the inverse is hard to compute.…
We establish several relations between quantum error correction (QEC) and tensor network (TN) methods of quantum many-body physics. We exhibit correspondences between well-known families of QEC codes and TNs, and demonstrate a formal equivalence between decoding a QEC code and contracting a TN. We build on this equivalence to propose a new family of quantum codes and decoding algorithms that generalize and improve upon quantum polar codes and successive cancellation decoding in a natural way.
Sampling from the lattice Gaussian distribution plays an important role in various research fields. In this paper, the Markov chain Monte Carlo (MCMC)-based sampling technique is advanced in several fronts. Firstly, the spectral gap for the independent Metropolis-Hastings-Klein (MHK) algorithm is derived, which is then extended to Peikert's algorithm and rejection sampling; we show that independent MHK exhibits faster convergence. Then, the performance of bounded distance decoding using MCMC is analyzed, revealing a flexible trade-off between the decoding radius and complexity. MCMC is further…
For communication systems with heavy burst noise, an optimal Forward Error Correction (FEC) scheme is expected to have a large burst error correction capacity while simultaneously owning moderate random error correction capability. This letter presents a new FEC scheme based on multiple-symbol interleaved Reed-Solomon codes and an associated two-pass decoding algorithm. It is shown that the proposed multi-symbol interleaved coding scheme can achieve nearly twice as much as the burst error correction capability of conventional symbol-interleaved Reed-Solomon codes with the same code length and …
We correct errors in Section III-B of the above-titled paper. The corrections are related to the joint distribution of the inputs to the main decoder which corresponds to a branch in the code trellis for the main decoder. Note that as far as the distribution of the input corresponding to a single code symbol is concerned, the results in Section III-B are correct. We have noticed that alpha in Proposition 12 (beta in Proposition 14) has another meaning, which has made it possible to derive the joint distribution of the inputs corresponding to the whole branch. Corrections are based on this impo…
A novel permutation decoding method for Reed-Muller codes is presented. The complexity and the error correction performance of the suggested permutation decoding approach are similar to that of the recursive lists decoder. It is demonstrated that the proposed decoding technique can take advantage of several early termination methods leading to a significant reduction of the operations number required for the decoding, with the error correction performance being the same.
This work introduces a decoding strategy for binary self-dual codes possessing an automorphism of a specific type. The proposed algorithm is a hard decision iterative decoding scheme. The enclosed experiments show that the new decoding concept performs error correction beyond the upper bound for the code correction capability. Moreover, we prove that the requirements for the new algorithm hold for any binary self-dual code having an automorphism of the specific type, which makes decoding of this large group of codes possible.
Quantum errors are primarily detected and corrected using the measurement of syndrome information which itself is an unreliable step in practical error correction implementations. Typically, such faulty or noisy syndrome measurements are modeled as a binary measurement outcome flipped with some probability. However, the measured syndrome is in fact a discretized value of the continuous voltage or current values obtained in the physical implementation of the syndrome extraction. In this paper, we use this "soft" or analog information without the conventional discretization step to benefit the i…
We consider transmission over a general memoryless channel, with bounded decoding complexity per bit under message passing decoding. We show that the achievable rate is bounded below capacity if there is a finite success in the decoding in a specified number of operations per bit at the decoder for some codes on graphs. These codes include LDPC and LDGM codes. Good performance with low decoding complexity suggests strong local structures in the graphs of these codes, which are detrimental to the code rate asymptotically. The proof method leads to an interesting necessary condition on the code …
Dynamic successive cancellation flip (DSCF) decoding of polar codes is a powerful algorithm that can achieve the error correction performance of successive cancellation list (SCL) decoding, with a complexity that is close to that of successive cancellation (SC) decoding at practical signal-to-noise ratio (SNR) regimes. However, DSCF decoding requires costly transcendental computations which adversely affect its implementation complexity. In this paper, we first show that a direct application of common approximation schemes on the conventional DSCF decoding results in significant error-correcti…
We revisit the error correction scheme of real-valued signals when the codeword is corrupted by gross errors on a fraction of entries and a small noise on all the entries. Combining the recent developments of approximate message passing and the spatially-coupled measurement matrix in compressed sensing we show that the error correction and its robustness towards noise can be enhanced considerably. We discuss the performance in the large signal limit using previous results on state evolution, as well as for finite size signals through numerical simulations. Even for relatively small sizes, the …
Despite its reduced complexity, lattice reduction-aided decoding exhibits a widening gap to maximum-likelihood (ML) performance as the dimension increases. To improve its performance, this paper presents randomized lattice decoding based on Klein's sampling technique, which is a randomized version of Babai's nearest plane algorithm (i.e., successive interference cancelation (SIC)). To find the closest lattice point, Klein's algorithm is used to sample some lattice points and the closest among those samples is chosen. Lattice reduction increases the probability of finding the closest lattice po…
We continue the study of rateless codes for transmission of information across channels whose rate of erasure is unknown. In such a code, an infinite stream of encoding symbols can be generated from the message and sent across the erasure channel, and the decoder can decode the message after it has successfully collected a certain number of encoding symbols. A rateless erasure code is real-time oblivious if rather than collecting encoding symbols as they are received, the receiver either immediately decodes or discards each symbol it receives. Efficient real-time oblivious erasure correction u…
A generic $(r,m)$-erasure correcting set is a collection of vectors in $\bF_2^r$ which can be used to generate, for each binary linear code of codimension $r$, a collection of parity check equations that enables iterative decoding of all correctable erasure patterns of size at most $m$. That is to say, the only stopping sets of size at most $m$ for the generated parity check equations are the erasure patterns for which there is more than one manner to fill in theerasures to obtain a codeword. We give an explicit construction of generic $(r,m)$-erasure correcting sets of cardinality $\sum_{i=0}…
Error correction codes are an integral part of communication applications, boosting the reliability of transmission. The optimal decoding of transmitted codewords is the maximum likelihood rule, which is NP-hard due to the curse of dimensionality. For practical realizations, sub-optimal decoding algorithms are employed; yet limited theoretical insights prevent one from exploiting the full potential of these algorithms. One such insight is the choice of permutation in permutation decoding. We present a data-driven framework for permutation selection, combining domain knowledge with machine lear…
Iterative Bounded Distance Decoding With 2-Bit Reliability Marker For Product-Like Codes
Technology and Engineering Center for Space Utilization, Chinese Academy of Sciences（中国科学院空间应用工程与技术中心）；Global College, Shanghai Jiao Tong University（上海交通大学全球学院）
Iterative decoders used for decoding low-density parity-check (LDPC) and moderate-density parity-check (MDPC) codes are not characterized by a deterministic decoding radius and their error rate performance is usually assessed through intensive Monte Carlo simulations. However, several applications, like code-based cryptography, need guaranteed low values of the error rate, which are infeasible to assess through simulations, thus requiring the development of theoretical models for the error rate of these codes under iterative decoding. Some models of this type already exist, but become computat…
In this paper we present an alternative separable implementation of L2-orthogonal space-time codes (STC) for continuous phase modulation (CPM). In this approach, we split the STC CPM transmitter into a single conventional CPM modulator and a correction filter bank. While the CPM modulator is common to all transmit antennas, the correction filter bank applies different correction units to each antenna. Thereby desirable code properties as orthogonality and full diversity are achievable with just a slightly larger bandwidth demand. This new representation has three main advantages. First, it all…
Storage systems have a strong need for substantially improving their error correction capabilities, especially for long-term storage where the accumulating errors can exceed the decoding threshold of error-correcting codes (ECCs). In this work, a new scheme is presented that uses deep learning to perform soft decoding for noisy files based on their natural redundancy. The soft decoding result is then combined with ECCs for substantially better error correction performance. The scheme is representation-oblivious: it requires no prior knowledge on how data are represented (e.g., mapped from symb…
In this paper we propose a very simple but powerful self-correction method for the Min-Sum decoding of LPDC codes. Unlike other correction methods known in the literature, our method does not try to correct the check node processing approximation, but it modifies the variable node processing by erasing unreliable messages. However, this positively affects check node messages, which become symmetric Gaussian distributed, and we show that this is sufficient to ensure a quasi-optimal decoding performance. Monte-Carlo simulations show that the proposed Self-Corrected Min-Sum decoding performs very…
Belief propagation is a powerful tool in statistical physics, machine learning, and modern coding theory. As a decoding method, it is ubiquitous in classical error correction and has also been applied to stabilizer-based quantum error correction. The algorithm works by passing messages between nodes of the factor graph associated with the code and enables efficient decoding, in some cases even up to the Shannon capacity of the channel. Here we construct a belief propagation algorithm which passes quantum messages on the factor graph and is capable of decoding the classical-quantum channel with…
In this work we present some results that allow to improve the decoding radius in solving polynomial linear systems with errors in the scenario where errors are additive and randomly distributed over a finite field. The decoding radius depends on some bounds on the solution that we want to recover, so their overestimation could significantly decrease our error correction capability. For this reason, we introduce an algorithm that can bridge this gap, introducing some ad hoc parameters that reduce the discrepancy between the estimate decoding radius and the effective error correction capability…
In this paper we present an FPGA-based implementation of linear programming (LP) decoding. LP decoding frames error correction as an optimization problem. This is in contrast to variants of belief propagation (BP) decoding that view error correction as a problem of graphical inference. There are many advantages to taking the optimization perspective: convergence guarantees, improved performance in certain regimes, and a methodology for incorporating the latest developments in optimization techniques. However, LP decoding, when implemented with standard LP solvers, does not easily scale to the …
Polar codes are a channel coding scheme for the next generation of wireless communications standard (5G). The belief propagation (BP) decoder allows for parallel decoding of polar codes, making it suitable for high throughput applications. However, the error-correction performance of polar codes under BP decoding is far from the requirements of 5G. It has been shown that the error-correction performance of BP can be improved if the decoding is performed on multiple permuted factor graphs of polar codes. However, a different BP decoding scheduling is required for each factor graph permutation w…
Reed-Muller (RM) and polar codes are a class of capacity-achieving channel coding schemes with the same factor graph representation. Low-complexity decoding algorithms fall short in providing a good error-correction performance for RM and polar codes. Using the symmetric group of RM and polar codes, the specific decoding algorithm can be carried out on multiple permutations of the factor graph to boost the error-correction performance. However, this approach results in high decoding complexity. In this paper, we first derive the total number of factor graph permutations on which the decoding c…
An unknown vector f in R^n can be recovered from corrupted measurements y = Af + e where A^(m*n)(m>n) is the coding matrix if the unknown error vector e is sparse. We investigate the relationship of the fraction of errors and the recovering ability of lp-minimization (0 < p <= 1) which returns a vector x minimizing the "lp-norm" of y - Ax. We give sharp thresholds of the fraction of errors that determine the successful recovery of f. If e is an arbitrary unknown vector, the threshold strictly decreases from 0.5 to 0.239 as p increases from 0 to 1. If e has fixed support and fixed signs on the …
The interest in polar codes has been increasing significantly since their adoption for use in the 5$^{\rm th}$ generation wireless systems standard. Successive cancellation (SC) decoding algorithm has low implementation complexity, but yields mediocre error-correction performance at the code lengths of interest. SC-Flip algorithm improves the error-correction performance of SC by identifying possibly erroneous decisions made by SC and re-iterates after flipping one bit. It was recently shown that only a portion of bit-channels are most likely to be in error. In this work, we investigate the av…
The 5th generation wireless systems (5G) standardization process of the 3rd generation partnership project (3GPP) chose polar codes as a channel coding scheme for the control channel. In case of downlink control information, polar codes are concatenated with distributed distributed cyclic redundancy check (CRC). Whereas CRC bits allow to improve the performance of successive cancellation list (SCL) decoders by improving distance properties, distributed CRC bits allow for path pruning and decoding early-termination. In this paper, we show how to take advantage of the distributed CRC to improve …
The quantum error correction theory is as a rule formulated in a rather convoluted way, in comparison to classical algebraic theory. This work revisits the error correction in a noisy quantum channel so as to make it intelligible to engineers. An illustrative example is presented of a naive perfect quantum code (Hamming-like code) with five-qubits for transmitting a single qubit of information. Also the (9,1)-Shor codes is addressed.
Polar codes are a class of linear block codes that provably achieves channel capacity, and have been selected as a coding scheme for $5^{\rm th}$ generation wireless communication standards. Successive-cancellation (SC) decoding of polar codes has mediocre error-correction performance on short to moderate codeword lengths: the SC-Flip decoding algorithm is one of the solutions that have been proposed to overcome this issue. On the other hand, SC-Flip has a higher implementation complexity compared to SC due to the required log-likelihood ratio (LLR) selection and sorting process. Moreover, it …
Extensive quantum error correction is necessary in order to scale quantum hardware to the regime of practical applications. As a result, a significant amount of decoding hardware is necessary to process the colossal amount of data required to constantly detect and correct errors occurring over the millions of physical qubits driving the computation. The implementation of a recent highly optimized version of Shor's algorithm to factor a 2,048-bits integer would require more 7 TBit/s of bandwidth for the sole purpose of quantum error correction and up to 20,000 decoding units. To reduce the deco…
Polar codes are capacity achieving error correcting codes that can be decoded through the successive-cancellation algorithm. To improve its error-correction performance, a list-based version called successive-cancellation list (SCL) has been proposed in the past, that however substantially increases the number of time-steps in the decoding process. The simplified SCL (SSCL) decoding algorithm exploits constituent codes within the polar code structure to greatly reduce the required number of time-steps without introducing any error-correction performance loss. In this paper, we propose a faster…
We present error-correcting codes that achieve the information-theoretically best possible trade-off between the rate and error-correction radius. Specifically, for every $0 < R < 1$ and $\eps> 0$, we present an explicit construction of error-correcting codes of rate $R$ that can be list decoded in polynomial time up to a fraction $(1-R-\eps)$ of {\em worst-case} errors. At least theoretically, this meets one of the central challenges in algorithmic coding theory. Our codes are simple to describe: they are {\em folded Reed-Solomon codes}, which are in fact {\em exactly} Reed-Solomon (RS) codes…
In this paper we propose an enhanced soft cancellation (SCAN) decoder for polar codes based on decoding stages permutation. The proposed soft cancellation list (SCANL) decoder runs $L$ independent SCAN decoders, each one relying on a different permuted factor graph. The estimated bits are selected among the $L$ candidates through a dedicated metric provided by the decoders. Furthermore, we introduce an early-termination scheme reducing decoding latency without affecting error correction performance. We investigate the error-correction performance of the proposed scheme under various combinatio…
A new permutation decoding approach for polar codes is presented. The complexity of the algorithm is similar to that of a successive cancellation list (SCL) decoder, while it can be implemented with the latency of a successive cancellation decoder. As opposed to the SCL algorithm, the sorting operation is not used in the proposed method. It is shown that the error correction performance of the algorithm is similar to that of the SCL decoder for polar codes. Moreover, a new construction aiming to improve the error correction performance of polar codes under the proposed algorithm is presented.
Polar codes are a class of channel capacity achieving codes that has been selected for the next generation of wireless communication standards. Successive-cancellation (SC) is the first proposed decoding algorithm, suffering from mediocre error-correction performance at moderate code length. In order to improve the error-correction performance of SC, two approaches are available: (i) SC-List decoding which keeps a list of candidates by running a number of SC decoders in parallel, thus increasing the implementation complexity, and (ii) SC-Flip decoding that relies on a single SC module, and kee…
Despite the extreme error-correction performance, the amount of computation of sequential decoding of the polarization-adjusted convolutional (PAC) codes is random. In sequential decoding of convolutional codes, the computational cutoff rate denotes the region between rates whose average computational complexity of decoding is finite and those which is infinite. In this paper, by benefiting from the polarization and guessing techniques, we prove that the computational cutoff rate in sequential decoding of pre-transformed polar codes polarizes. The polarization of the computational cutoff rate …
Research on polar codes has been constantly gaining attention over the last decade, by academia and industry alike, thanks to their capacity-achieving error-correction performance and low-complexity decoding algorithms. Recently, they have been selected as one of the coding schemes in the $5^{th}$ generation wireless standard (5G). Over the years various polar code decoding algorithms, like SC-list (SCL), have been proposed to improve the mediocre performance of the successive cancellation (SC) decoding algorithm for finite code lengths; however, like SC, they suffer from long decoding latency…
Product codes are widespread in optical communications, thanks to their high throughput and good error-correction performance. Systematic polar codes have been recently considered as component codes for product codes. In this paper, we present a novel construction for product polar codes based on non-systematic polar codes. We prove that the resulting product code is actually a polar code, having a frozen set that is dependent on the frozen sets of the component polar codes. We propose a low-complexity decoding algorithm exploiting the dual nature of the constructed code. Performance analysis …
In this paper, we present an optimal metric function on average, which leads to a significantly low decoding computation while maintaining the superiority of the polarization-adjusted convolutional (PAC) codes' error-correction performance. With our proposed metric function, the PAC codes' decoding computation is comparable to the conventional convolutional codes (CC) sequential decoding. Moreover, simulation results show an improvement in the low-rate PAC codes' error-correction performance when using our proposed metric function. We prove that choosing the polarized cutoff rate as the metric…
Undetected errors are important for linear codes, which are the only type of errors after hard decision and automatic-repeat-request (ARQ), but do not receive much attention on their correction. In concatenated channel coding, suboptimal source coding and joint source-channel coding, constrains among successive codewords may be utilized to improve decoding performance. In this paper, list decoding is used to correct the undetected errors. The benefit proportion of the correction is obviously improved especially on Hamming codes and Reed-Muller codes, which achieves about 40% in some cases. But…
