Decoding Of Codes Research Articles

List-Based Optimization of Proximal Decoding for LDPC Codes

List-Based Optimization of Proximal Decoding for LDPC Codes

Hardness results for decoding the surface code with Pauli noise

Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit.In this setting, we show that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for QMLD and DQMLD. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for QMLD and DQMLD for arbitrary stabilizer codes with independent X and Z noise.

Two-Phase Globally Coupled Low-Density Parity Check Decoding Aided with Early Termination and Forced Convergence.

To enhance the decoding efficiency of Globally Coupled (GC) LDPC codes, we incorporated Early Termination (ET) and Forced Convergence (FC) into the local/global two-phase decoding algorithm to expedite the decoding process. The two-phase decoding scheme integrates the ET technique to halt unnecessary iterations in the local decoding phase while employing the FC technique to accelerate convergence in the global phase decoding. The application of ET technology in the local decoding of GC-LDPC codes will not cause performance loss as in traditional block codes and will cause considerable complexity gains. For a longer code length and larger convergence differences between nodes' global codes, the FC technique operates more efficiently in global code than local code. Two variants are proposed for the ET scheme in the local decoding, namely ET-1 and ET-2. The initial variant, ET-1, predicts whether local decoding can be successful according to data characteristics and stop the local decoding iteration that is not expected to be successful in time. In the case of ET-2, the saved local iterations are transformed to global decoding equally. The results show that ET-1 saves considerable decoding time complexity and ET-2 improves the performance of the GC-LDPC code with the same decoding time complexity. The combined approach of ET-1 with FC reduces the decoding time complexity up to 42% at a low Signal Noise Rate region while maintaining its performance; ET-2-FC two-phase decoding saves approximately 25% decoding time complexity while improving the BER by about 0.18 dB and FER by about 0.23 dB.

Decoding quantum color codes with MaxSAT

In classical computing, error-correcting codes are well established and are ubiquitous both in theory and practical applications. For quantum computing, error-correction is essential as well, but harder to realize, coming along with substantial resource overheads and being concomitant with needs for substantial classical computing. Quantum error-correcting codes play a central role on the avenue towards fault-tolerant quantum computation beyond presumed near-term applications. Among those, color codes constitute a particularly important class of quantum codes that have gained interest in recent years due to favourable properties over other codes. As in classical computing, decoding is the problem of inferring an operation to restore an uncorrupted state from a corrupted one and is central in the development of fault-tolerant quantum devices. In this work, we show how the decoding problem for color codes can be reduced to a slight variation of the well-known LightsOut puzzle. We propose a novel decoder for quantum color codes using a formulation as a MaxSAT problem based on this analogy. Furthermore, we optimize the MaxSAT construction and show numerically that the decoding performance of the proposed decoder achieves state-of-the-art decoding performance on color codes. The implementation of the decoder as well as tools to automatically conduct numerical experiments are publicly available as part of the Munich Quantum Toolkit (MQT) on GitHub.

Accurate optimal quantum error correction thresholds from coherent information

Quantum error correcting (QEC) codes protect quantum information from decoherence as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, suboptimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered, we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close agreement with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models. Published by the American Physical Society 2024

Optimization of Decoder Priors for Accurate Quantum Error Correction.

Accurate decoding of quantum error-correcting codes is a crucial ingredient in protecting quantum information from decoherence. It requires characterizing the error channels corrupting the logical quantum state and providing this information as a prior to the decoder. We introduce a reinforcement learning inspired method for calibrating these priors that aims to minimize the logical error rate. Our method significantly improves the decoding accuracy in repetition and surface code memory experiments executed on Google's Sycamore processor, outperforming the leading decoder-agnostic method by 16% and 3.3%, respectively. This calibration approach will serve as an important tool for maximizing the performance of both near-term and future error-corrected quantum devices.

Hybrid Decoding of Asymmetric Product Codes With Systematic Polar and BCH Codes

Hybrid Decoding of Asymmetric Product Codes With Systematic Polar and BCH Codes

SIGACT News Complexity Theory Column

This month's guest author is Dean Doron, who provides a wonderful survey on construction, encoding and decoding of binary codes with distance close to half. Thank you Dean for your hard work! Coming up next, we'll have columns by Stacey Jeffrey (on Quantum Las Vegas Algorithms and Composition), Igor Carboni Oliveira (on the Meta-Mathematics of Computational Complexity Theory), Noam Mazor (topic TBA) and Elena Grigorescu (on Locally Decodable Codes for Insertions and Deletions). Stay tuned for more!

A path splitting and pruning strategy on list decoder for PAC codes

AbstractRecently, Arıkan proposed a polarization‐adjusted convolutional (PAC) codes, demonstrating their superior error correction performance over polar codes at short block lengths. It was confirmed that PAC codes approached the optimal performance achievable with limited code length. This paper proposes a novel low‐complexity list decoding algorithm for PAC codes, incorporating path splitting and pruning strategies based on a set of highly reliable information bits. Simulation results reveal that the proposed algorithm significantly reduces sorting complexity and average list size, all while incurring negligible performance loss. Unlike previous pruning algorithms designed for polar codes, the proposed strategy eliminates the need to individually assess the reliability of decoding paths in each decoding process. Instead, the algorithm minimizes redundant decoding paths through a high‐reliability information bit set, constructed using Monte Carlo experiments.

Fast erasure decoder for hypergraph product codes

We propose a decoder for the correction of erasures with hypergraph product codes, which form one of the most popular families of quantum LDPC codes. Our numerical simulations show that this decoder provides a close approximation of the maximum likelihood decoder that can be implemented in O(N2) bit operations where N is the length of the quantum code. A probabilistic version of this decoder can be implemented in O(N1.5) bit operations.

Matrix hypercontractivity, streaming algorithms and LDCs: the large alphabet case

We prove a hypercontractive inequality for matrix-valued functions defined over large alphabets. In order to do so, we prove a generalization of the powerful 2-uniform convexity inequality for trace norms of Ball, Carlen, Lieb (Inventiones Mathematicae’94). Using our hypercontractive inequality, we present upper and lower bounds for the communication complexity of the Hidden Hypermatching problem defined over large alphabets. We then consider streaming algorithms for approximating the value of Unique Games on a hypergraph with t -size hyperedges. By using our communication lower bound, we show that every streaming algorithm in the adversarial model achieving an ( r − ε)-approximation of this value requires Ω ( n 1 − 2/ t ) quantum space, where r is the alphabet size. We next present a lower bound for locally decodable codes ( \(\mathsf {LDC} \) ) \(\mathbb {Z}_r^n\rightarrow \mathbb {Z}_r^N \) over large alphabets with recoverability probability at least 1/ r + ε. Using hypercontractivity, we give an exponential lower bound \(N= 2^{\Omega (\varepsilon ^4 n/r^4)} \) for 2-query (possibly non-linear) \(\mathsf {LDC} \) s over \(\mathbb {Z}_r \) and using the non-commutative Khintchine inequality we prove an improved lower bound of \(N= 2^{\Omega (\varepsilon ^2 n/r^2)} \) .

An Investigation on LP Decoding of Short Binary Linear Codes With the Subgradient Method

An Investigation on LP Decoding of Short Binary Linear Codes With the Subgradient Method

Triangle Projection Algorithm in ADMM-LP Decoding of LDPC Codes

Triangle Projection Algorithm in ADMM-LP Decoding of LDPC Codes

Efficient turbo product code decoder with Build‐In SRAM‐based transpose memory

AbstractTurbo product codes (TPCs) have been widely used for bit error correction in high‐speed applications such as data storage. This letter introduces an efficient hard‐input hard‐output iterating TPC decoder module. A transpose memory utilizing static random access memory (SRAM) is integrated into the decoder to achieve a low hardware overhead. The transpose memory, based on an 8T SRAM bit‐cell, supports both horizontal (row‐wise), and vertical (column‐wise) read/write operations. It is prototyped under a 28nm high‐k/metal gate stack process with bit‐cell size of 0.582 µm2. This specialized SRAM significantly reduces the hardware overhead when compared with a register‐array‐based transpose memory without significant throughput loss. A field programmable gate array (FPGA) evaluation platform is utilized to emulate the TPC decoder module, and the maximum decoder throughput is up to 6.49 Gbps at 250 MHz.

Decoding of MDP Convolutional Codes over the Erasure Channel under Linear Systems Point of View

This paper attempts to highlight the decoding capabilities of MDP convolutional codes over the erasure channel by defining them as discrete linear dynamical systems, with which the controllability property and the observability characteristics of linear system theory can be applied, in particular those of output observability, easily described using matrix language. Those are viewed against the decoding capabilities of MDS block codes over the same channel. Not only is the time complexity better but the decoding capabilities are also increased with this approach because convolutional codes are more flexible in handling variable-length data streams than block codes, where they are fixed-length and less adaptable to varying data lengths without padding or other adjustments.

Design of DNA Storage Coding Scheme With LDPC Codes and Interleaving.

In this paper, we propose a new coding scheme for DNA storage using low-density parity-check (LDPC) codes and interleaving techniques. While conventional coding schemes generally employ error correcting codes in both inter and intra-oligo directions, we show that inter-oligo LDPC codes, optimized by differential evolution, are sufficient in ensuring the reliability of DNA storage due to the powerful soft decoding of LDPC codes. In addition, we apply interleaving techniques for handling non-uniform error characteristics of DNA storage to enhance the decoding performance. Consequently, the proposed coding scheme reduces the required number of oligo reads for perfect recovery by 26.25% ~ 38.5% compared to existing state-of-the-art coding schemes. Moreover, we develop an analytical DNA channel model in terms of non-uniform binary symmetric channels. This mathematical model allows us to demonstrate the superiority of the proposed coding scheme while isolating the experimental variation, as well as confirm the independent effects of LDPC codes and interleaving techniques.

Implementation of turbo decoding for VDES signal data segments

Abstract The International Telecommunication Union (ITU) mandates the use of turbo coding for VDES signal data segments. Previous studies primarily focused on turbo decoding for a specific link in VDES, often neglecting its applicability to all links. This paper proposes an FPGA-based implementation method for turbo code decoding, aiming to achieve fast and accurate turbo decoding of VDES signal data segments, while also ensuring compatibility with different links. The resulting turbo decoder can adapt to the different code rates and data segment lengths specified in the VDES. A comparison of the decoding speeds between DSP and FPGA confirms that the parallel computing capability of FPGA can significantly improve the decoding speed. Finally, the main onboard resources required for FPGA implementation of turbo decoding were provided.

Fast Decoding of Polar Codes for Digital Broadcasting Services in 5G

Fast Decoding of Polar Codes for Digital Broadcasting Services in 5G

Optimized decoder for low-density parity check codes based on genetic algorithms

Low-density parity check (LDPC) codes, are a family of error-correcting codes, their performances close to the Shannon limit make them very attractive solutions for digital communication systems. There are several algorithms for decoding LDPC codes that show great diversity in terms of performance related to error correction. Also, very recently, many research papers involved the genetic algorithm (GA) in coding theory, in particular, in the decoding linear block codes case, which has heavily contributed to reducing the bit error rate (BER). In this paper, an efficient method based on the GA is proposed and it is used to improve the power of correction in terms of BER and the frame error rate (FER) of LDPC codes. Subsequently, the proposed algorithm can independently decide the most suitable moment to stop the decoding process, moreover, it does not require channel information (CSI) making it adaptable for all types of channels with different noise or intensity. The simulations show that the proposed algorithm is more efficient in terms of BER compared to other LDPC code decoders.

Advanced channel coding schemes for B5G/6G networks: State‐of‐the‐art analysis, research challenges and future directions

SummaryAs a consequence of continuing demand for additional capacity and higher quality data transmission, channel coding as a key component of a digital communication network has progressed from a classical single pair, point‐to‐point information theoretic application to a class of coding techniques with diverse parameters. The heterogeneous requirements of B5G/6G networks technology have made flexibility of code parameters the main desired feature while designing channel codes. Owing to this, channel coding techniques have evolved to support enabling applications that depend on different factors such as latency, reliability, energy efficiency and throughput. We review and discuss the application of new techniques, modifications in the design and construction of modern channel coding schemes, including block, convolutional, rateless and space‐time codes. Further, the error correction performance of mainstream channel decoders for LDPC, polar and turbo codes is compared and analysed for their attainable error rate. The aim of this study is to highlight the adaptability of the discussed coding schemes for the forthcoming cellular network landscape. This article also addresses the implementational challenges of high throughput, low latency and machine type communication scenarios. Moreover, some recent studies exploring the role of machine learning for optimisation of B5G/6G networks are classified and discussed. Concluding, research areas pertaining to the scalability of error control coding for next generation networking are addressed.