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Optimal Codes for the Burst Erasure Channel
Deep space communications over noisy channels lead to certain packets that are not decodable. These packets leave gaps, or bursts of erasures, in the data stream. Burst erasure correcting codes overcome this problem. These are forward erasure correcting codes that allow one to recover the missing gaps of data. Much of the recent work on this topic concentrated on Low-Density Parity-Check (LDPC) codes. These are more complicated to encode and decode than Single Parity Check (SPC) codes or Reed-Solomon (RS) codes, and so far have not been able to achieve the theoretical limit for burst erasure protection. A block interleaved maximum distance separable (MDS) code (e.g., an SPC or RS code) offers near-optimal burst erasure protection, in the sense that no other scheme of equal total transmission length and code rate could improve the guaranteed correctible burst erasure length by more than one symbol. The optimality does not depend on the length of the code, i.e., a short MDS code block interleaved to a given length would perform as well as a longer MDS code interleaved to the same overall length. As a result, this approach offers lower decoding complexity with better burst erasure protection compared to other recent designs for the burst erasure channel (e.g., LDPC codes). A limitation of the design is its lack of robustness to channels that have impairments other than burst erasures (e.g., additive white Gaussian noise), making its application best suited for correcting data erasures in layers above the physical layer. The efficiency of a burst erasure code is the length of its burst erasure correction capability divided by the theoretical upper limit on this length. The inefficiency is one minus the efficiency. The illustration compares the inefficiency of interleaved RS codes to Quasi-Cyclic (QC) LDPC codes, Euclidean Geometry (EG) LDPC codes, extended Irregular Repeat Accumulate (eIRA) codes, array codes, and random LDPC codes previously proposed for burst erasure protection. As can be seen, the simple interleaved RS codes have substantially lower inefficiency over a wide range of transmission lengths.
MLEC-Sim: A Simulator for Evaluating Multi-Level Erasure Coding
We present MLEC-Sim, a sophisticated simulator for Multi-Level Erasure Coding (MLEC), developed in approximately 13 KLOC. The simulator is engineered to analyze the impact of various system configurations and erasure coding policies on system durability and network overhead. It supports a comprehensive range of parameters including disk capacity, disk I/O bandwidth, failure rates, network bandwidth, and system scale, accommodating various erasure coding approaches such as Single-Level Erasure Coding (SLEC), Multi-Level Erasure Coding (MLEC), and Local Reconstruction Codes (LRC). MLEC-Sim provides support for multiple chunk placement policies, including clustered parity and declustered parity, and encompasses a variety of repair methods like Repair-ALL, Repair-FCO, Repair-HYB, and Repair-MIN. It is capable of simulating disk failures through a variety of means, including distribution-based or trace-based mechanisms, and can handle complex multi-level (de)clustered placements and repair processes. A key feature of MLEC-Sim is its adoption of the splitting simulation method for evaluating system durabilities at extremely high levels, which are challenging to assess with traditional simulation approaches. This feature allows for a detailed evaluation of system resilience under a range of conditions, aiding in the selection of appropriate erasure coding solutions for enhancing system durability. MLEC-Sim contributes to the field of data storage and reliability by providing a tool for the detailed evaluation of the durability and efficiency of erasure coding configurations, intended for use by researchers and practitioners in the design and optimization of storage systems.
Erasure conversion in a high-fidelity Rydberg quantum simulator
Minimizing and understanding errors is critical for quantum science, both in noisy intermediate scale quantum (NISQ) devices and for the quest towards fault-tolerant quantum computation. Rydberg arrays have emerged as a prominent platform in this context with impressive system sizes and proposals suggesting how error-correction thresholds could be significantly improved by detecting leakage errors with single-atom resolution, a form of erasure error conversion. However, two-qubit entanglement fidelities in Rydberg atom arrays have lagged behind competitors and this type of erasure conversion is yet to be realized for matter-based qubits in general. Here we demonstrate both erasure conversion and high-fidelity Bell state generation using a Rydberg quantum simulator. When excising data with erasure errors observed via fast imaging of alkaline-earth atoms, we achieve a Bell state fidelity of $\ge 0.997{1}_{-13}^{+10}$ , which improves to $\ge 0.998{5}_{-12}^{+7}$ when correcting for remaining state-preparation errors. We further apply erasure conversion in a quantum simulation experiment for quasi-adiabatic preparation of long-range order across a quantum phase transition, and reveal the otherwise hidden impact of these errors on the simulation outcome. Our work demonstrates the capability for Rydberg-based entanglement to reach fidelities in the 0.999 regime, with higher fidelities a question of technical improvements, and shows how erasure conversion can be utilized in NISQ devices. These techniques could be translated directly to quantum-error-correction codes with the addition of long-lived qubits.
Nonequilibrium formulation of varying-temperature bit erasure
Landauer's principle states that erasing a bit of information at fixed temperature T costs at least $k$ B $T$ ln 2 units of work. Here we investigate erasure at varying temperature, to which Landauer's result does not apply. Here we formulate bit erasure as a stochastic nonequilibrium process involving a compression of configuration space, with physical and logical states associated in a symmetric way. Erasure starts and ends at temperature T, but temperature can otherwise vary with time in an arbitrary way. Defined in this way, erasure is governed by a set of nonequilibrium fluctuation relations that show that varying-temperature erasure can done with less work than $k$ B $T$ ln 2. As a result, erasure and the complementary process of bit randomization can be combined to form a work-producing engine cycle.
Chiral charge erasure via thermal fluctuations of magnetic helicity
We consider a relativistic plasma of fermions coupled to an Abelian gauge field and carrying a chiral charge asymmetry, which might arise in the early Universe through baryogenesis. It is known that on large length scales, λ≳1/(αμ{sub 5}), the chiral anomaly opens an instability toward the erasure of chiral charge and growth of magnetic helicity. Here the chemical potential μ{sub 5} parametrizes the chiral asymmetry and α is the fine-structure constant. We study the process of chiral charge erasure through the thermal fluctuations of magnetic helicity and contrast with the well-studied phenomenon of Chern-Simons number diffusion. Through the fluctuation-dissipation theorem we estimate the amplitude and time scale of helicity fluctuations on the length scale λ, finding δ H∼λT and τ∼αλ{sup 3}T{sup 2} for a relativistic plasma at temperature T. We argue that the presence of a chiral asymmetry allows the helicity to grow diffusively for a time t∼T{sup 3}/(α{sup 5}μ{sub 5}{sup 4}) until it reaches an equilibrium value H∼μ{sub 5}T{sup 2}/α, and the chiral asymmetry is partially erased. If the chiral asymmetry is small, μ{sub 5}<T/α, this avenue for chiral charge erasure is found to be slower than the chiral magnetic effect for which t∼T/(α{sup 3}μ{sub 5}{sup 2}). This mechanism for chiral charge erasure can be important for the hypercharge sector of the Standard Model as well as extensions including U(1) gauge interactions, such as asymmetric dark matter models.
A simplified procedure for correcting both errors and erasures of a Reed-Solomon code using the Euclidean algorithm
It is well known that the Euclidean algorithm or its equivalent, continued fractions, can be used to find the error locator polynomial and the error evaluator polynomial in Berlekamp's key equation needed to decode a Reed-Solomon (RS) code. A simplified procedure is developed and proved to correct erasures as well as errors by replacing the initial condition of the Euclidean algorithm by the erasure locator polynomial and the Forney syndrome polynomial. By this means, the errata locator polynomial and the errata evaluator polynomial can be obtained, simultaneously and simply, by the Euclidean algorithm only. With this improved technique the complexity of time domain RS decoders for correcting both errors and erasures is reduced substantially from previous approaches. As a consequence, decoders for correcting both errors and erasures of RS codes can be made more modular, regular, simple, and naturally suitable for both VLSI and software implementation. An example illustrating this modified decoding procedure is given for a (15, 9) RS code.
Information transmission with continuous variable quantum erasure channels
Quantum capacity, as the key figure of merit for a given quantum channel, upper bounds the channel's ability in transmitting quantum information. Identifying different types of channels, evaluating the corresponding quantum capacity, and finding the capacity-approaching coding scheme are the major tasks in quantum communication theory. Quantum channel in discrete variables has been discussed enormously based on various error models, while error model in the continuous variable channel has been less studied due to the infinite dimensional problem. In this paper, we investigate a general continuous variable quantum erasure channel. By defining an effective subspace of the continuous variable system, we find a continuous variable random coding model. We then derive the quantum capacity of the continuous variable erasure channel in the framework of decoupling theory. The discussion in this paper fills the gap of a quantum erasure channel in continuous variable setting and sheds light on the understanding of other types of continuous variable quantum channels.
Investigation of the Use of Erasures in a Concatenated Coding Scheme
A new method for declaring erasures in a concatenated coding scheme is investigated. This method is used with the rate 1/2 K = 7 convolutional code and the (255, 223) Reed Solomon code. Errors and erasures Reed Solomon decoding is used. The erasure method proposed uses a soft output Viterbi algorithm and information provided by decoded Reed Solomon codewords in a deinterleaving frame. The results show that a gain of 0.3 dB is possible using a minimum amount of decoding trials.
Learning efficient erasure protocols for an underdamped memory
Here we apply evolutionary reinforcement learning to a simulation model to identify efficient time-dependent erasure protocols for a physical realization of a 1-bit memory using an underdamped mechanical cantilever. We show that these protocols, when applied to the cantilever in the laboratory, are considerably more efficient than our best hand-designed protocols. The learned protocols allow reliable high-speed erasure by minimizing the heating of the memory during its operation. More generally, the combination of methods used here opens the door to the rational design of efficient protocols for various physics applications.
Circuit-Based Leakage-to-Erasure Conversion in a Neutral-Atom Quantum Processor
Atom-loss errors are a major limitation of current state-of-the-art neutral-atom quantum computers and pose a significant challenge for scalable systems. In a quantum processor with cesium atoms, we demonstrate proof-of-principle circuit-based conversion of this form of leakage error to erasure errors via leakage-detection units (LDUs), which nondestructively map information about the presence or absence of the qubit onto the state of an ancilla. We benchmark the performance of the LDU using a three-outcome low-loss state-detection method and find that the LDU detects atom-loss errors with approximately 93.4% accuracy, limited by technical imperfections of our apparatus. We further compile and execute a SWAP LDU, wherein the roles of the original data atom and ancilla atom are exchanged under the action of the LDU, providing “free refilling” of atoms in the case of atom loss. This circuit-based leakage-to-erasure error conversion is a critical component of a neutral-atom quantum processor where the quantum information may significantly outlive the lifetime of any individual atom in the quantum register. Finally, we demonstrate that LDUs may also be used to handle other forms of leakage errors where population moves to states outside of the computational subspace.
Quantum erasure and the locality problem
The problem of locality arising in connection with the quantum-erasure experiments is considered using the algebraic approach. We demonstrate that, contrary to a widespread opinion, the results of these experiments can be reconciled with the existence of a local physical reality determining the results of local measurements. A clear physical interpretation of the quantum-erasure experiments, that is consistent with the classical concepts, is given.
Error-erasure decoding of product codes.
Two error-erasure decoding algorithms for product codes that correct all the error-erasure patterns guaranteed correctable by the minimum Hamming distance of the product code are given. The first algorithm works when at least one of the component codes is majority-logic decodable. The second algorithm works for any product code. Both algorithms use the decoders of the component codes.
Mechanical erasure of particle tracks - A tool for lunar microstratigraphic chronology.
Mechanical erasure of particle tracks caused by impact deformation is shown to be a common feature of lunar soil grains. Etching of lunar pyroxenes reveals deformation markings that are evidence of prior impact events. The soil microstratigraphic chronology can be derived if it is assumed that the minimum track density in a thin soil layer is commonly found in grains where such erasure was caused by the impact that deposited the soil. With Apollo 12 core samples as examples, the minimum track density found in each layer is used to calculate the longest exposure that layer could have received since it was last disturbed. By summing the upper limits on the exposure times of the different layers, a lower limit on the average deposition rate was estimated to be 0.35 cm/m.y.
Efficient program for decoding the /255, 223/ Reed-Solomon code over GF/2 to the 8th/ with both errors and erasures, using transform decoding
The paper deals with a method developed for decoding a (255, 223) Reed-Solomon code over GF(2 to the 8th) with both errors and erasures. The matrix of decoding times for correcting errors and erasures of the code using a simplified decoder is presented. It is shown that the algorithm proposed is faster by a factor of from three to seven.
Erasure decoding in burst-error channels
A proven means of communicating reliably in a burst-error channel is the code interleaving scheme. Code symbols from a number of component codes are interleaved before being sent through the channel. This method effectively distributes the error detection and correction burden among the component codes and makes errors occurring in a codeword from each component code more or less independent. Erasure decoding techniques allow further refinement on the code interleaving concept. Their application leads to improved overall code performance when the symbol depth of the lead code is shallow compared to the average error-burst length of the channel. Theoretical formulations derived for predicting the performance of separate decoding and erasure decoding schemes are valuable in providing reasonably good estimates on redundancy requirements of the component codes.
Optimum Code Rates for Noncoherent MFSK with Errors and Erasures Decoding over Rayleigh Fading Channels
In this paper, we analyze the performance of a communication system employing M-ary frequency shift keying (FSK) modulation with errors-and-erasures decoding using Viterbi ratio threshold technique for erasure insertion, in Rayleigh fading and AWGN channels.
Protograph LDPC Codes Over Burst Erasure Channels
In this paper we design high rate protograph based LDPC codes suitable for binary erasure channels. To simplify the encoder and decoder implementation for high data rate transmission, the structure of codes are based on protographs and circulants. These LDPC codes can improve data link and network layer protocols in support of communication networks. Two classes of codes were designed. One class is designed for large block sizes with an iterative decoding threshold that approaches capacity of binary erasure channels. The other class is designed for short block sizes based on maximizing minimum stopping set size. For high code rates and short blocks the second class outperforms the first class.