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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.
Random bit errors in pulse code modulated data during telemetry data processing
Error correcting binary block code decoding on Q-ary output channels /weighted erasure decoding/, considering applications
Efficient computation of the bit error rate (BER) for the coherent M-ary PSK signals with Gray code bit mapping is considered. A closed-form expression for the exact BER of 8-ary PSK is presented. Tight upper and lower bounds on BER are also obtained for M-ary PSK with larger M.
The Hubble Space Telescope Scientific Instrument Control and Data Handling Subsystem (SI C&DH) is designed to interface with five scientific instruments of the Space Telescope to provide ground and autonomous control and collect health and status information using the Standard Telemetry and Command Components (STACC) multiplex data bus. It also formats high throughput science data into packets. The packetized data is interleaved and Reed-Solomon encoded for error correction and Pseudo Random encoded. An inner convolutional coding with the outer Reed-Solomon coding provides excellent error correction capability. The subsystem is designed with the capacity for orbital replacement in order to meet a mission life of fifteen years. The spacecraft computer and the SI C&DH computer coordinate the activities of the spacecraft and the scientific instruments to achieve the mission objectives.
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.
Analytical approach for development burst error correction and detection cyclic codes does not require shortening techniques.
High-rate concatenated coding systems with bandwidth-efficient trellis inner codes and Reed-Solomon (RS) outer codes are investigated for application in high-speed satellite communication systems. Two concatenated coding schemes are proposed. In one the inner code is decoded with soft-decision Viterbi decoding, and the outer RS code performs error-correction-only decoding (decoding without side information). In the other, the inner code is decoded with a modified Viterbi algorithm, which produces reliability information along with the decoded output. In this algorithm, path metrics are used to estimate the entire information sequence, whereas branch metrics are used to provide reliability information on the decoded sequence. This information is used to erase unreliable bits in the decoded output. An errors-and-erasures RS decoder is then used for the outer code. The two schemes have been proposed for high-speed data communication on NASA satellite channels. The rates considered are at least double those used in current NASA systems, and the results indicate that high system reliability can still be achieved.
Physical and mathematical formulae and concepts are fundamental elements of scientific and engineering software. These classical equations and methods are time tested, universally accepted, and relatively unambiguous. The existence of this classical ontology suggests an ideal problem for automated comprehension. This problem is further motivated by the pervasive use of scientific code and high code development costs. To investigate code comprehension in this classical knowledge domain, a research prototype has been developed. The prototype incorporates scientific domain knowledge to recognize code properties (including units, physical, and mathematical quantity). Also, the procedure implements programming language semantics to propagate these properties through the code. This prototype's ability to elucidate code and detect errors will be demonstrated with state of the art scientific codes.
For rate 1/N convolutional codes, a recursive algorithm for finding the transfer function bound on bit error rate (BER) at the output of a Viterbi decoder is described. This technique is very fast and requires very little storage since all the unnecessary operations are eliminated. Using this technique, we find and plot bounds on the BER performance of known codes of rate 1/2 with K 18, rate 1/3 with K 14. When more than one reported code with the same parameter is known, we select the code that minimizes the required signal to noise ratio for a desired bit error rate of 0.000001. This criterion of determining goodness of a code had previously been found to be more useful than the maximum free distance criterion and was used in the code search procedures of very short constraint length codes. This very efficient technique can also be used for searches of longer constraint length codes.
Correction to: Naturehttps://doi.org/10.1038/s41586-024-07117-5 Published online 13 March 2024 In the version of the article initially published, several errors were present and have been corrected in the HTML and PDF versions of the article and Supplementary Information. The main results, conclusions, and our interpretations of the data remain unchanged. See the new Supplementary Information Section S15 for a more detailed description of the errors corrected and the resulting effects on the analysis. Data processing and methods corrections Overflight count correction: We previously used pre-computed source coverage data for some Carbon Mapper campaigns that was computed differently than was required for our analysis. We have re-computed Carbon Mapper source coverage based on flightline polygons and source coordinates. Transition point computation, well sites: The updated version now correctly compares the cumulative emissions distribution of simulated well site emissions with that of aerially detected sources (rather than plumes) when computing the transition point. Transition point computation, midstream: Additionally, the transition point calculation has been corrected to exclude aerially detected midstream emissions below the transition point, which was previously leading to double counting of these emissions. This error was not present for upstream (well site) emissions. Calculation errors Unit error: We corrected a specific unit conversion error affecting well site emissions in the Kairos Fort Worth dataset. Across all datasets, we also correct the conversion factor for converting from standard volume to mass for midstream emissions. Sorting error: We correct code that was applying incorrect sorting when computing correction factors to account for partial detection at well sites. Small typographical corrections were made in Fig. 1b and SI Section S4.1. Data processing and methods corrections Overflight count correction: We previously used pre-computed source coverage data for some Carbon Mapper campaigns that was computed differently than was required for our analysis. We have re-computed Carbon Mapper source coverage based on flightline polygons and source coordinates. Transition point computation, well sites: The updated version now correctly compares the cumulative emissions distribution of simulated well site emissions with that of aerially detected sources (rather than plumes) when computing the transition point. Transition point computation, midstream: Additionally, the transition point calculation has been corrected to exclude aerially detected midstream emissions below the transition point, which was previously leading to double counting of these emissions. This error was not present for upstream (well site) emissions. Calculation errors Unit error: We corrected a specific unit conversion error affecting well site emissions in the Kairos Fort Worth dataset. Across all datasets, we also correct the conversion factor for converting from standard volume to mass for midstream emissions. Sorting error: We correct code that was applying incorrect sorting when computing correction factors to account for partial detection at well sites. Small typographical corrections were made in Fig. 1b and SI Section S4.1. The following practices may help researchers conducting similar analyses avoid making similar errors: 1, Clear, accessible documentation explaining the interpretation of all columns in data input tables and all internal variables within the model, 2, Simple cross-check calculations computed before and after unit conversions.
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.
We report on the fault-tolerant operation of logical qubits on a neutral atom quantum computer, with logical performance surpassing physical performance for multiple circuits including Bell state preparation (12x error reduction), random circuits (15x), and a prototype Anderson Impurity Model ground state solver for materials science applications (up to 6x, non-fault-tolerantly). The logical qubits are implemented via the [[4, 2, 2]] code (C 4 ). Our work constitutes the first complete realization of the benchmarking protocol proposed by Gottesman 2016 demonstrating results consistent with fault tolerance. In light of recent advances on applying concatenated C 4 /C 6 detection codes to achieve error correction with high code rates and thresholds, our work can be regarded as a building block towards a practical scheme for fault tolerant quantum computation. Our demonstration of a materials science application with logical qubits particularly demonstrates the immediate value of these techniques on current experiments.
Recently several day and nighttime links under diverse atmospheric conditions were completed using the Optical Payload for Lasercomm Science (OPALS) flight system on-board the International Space Station (ISS). In this paper we compare measured optical power and its variance at either end of the link with predictions that include atmospheric propagation models. For the 976 nm laser beacon mean power transmitted from the ground to the ISS the predicted mean irradiance of 10's of microwatts per square meter close to zenith and its decrease with range and increased air mass shows good agreement with predictions. The irradiance fluctuations sampled at 100 Hz also follow the expected increase in scintillation with air mass representative of atmospheric coherence lengths at zenith at 500 nm in the 3-8 cm range. The downlink predicted power of 100's of nanowatts was also reconciled within the uncertainty of the atmospheric losses. Expected link performance with uncoded bit-error rates less than 1E-4 required for the Reed-Solomon code to correct errors for video, text and file transmission was verified. The results of predicted and measured powers and fluctuations suggest the need for further study and refinement.
A technique is presented for correction of an (n,k) cyclic block code subjected to a noise disturbance consisting of an arbitrary number of both bit deletions and bit inversions contained within a single error burst. Following the procedure described by Meggitt (1961), the correction of a b-bit burst is attempted by first loading the initial error syndrome into an (n-k) order feedback shift register with taps selected for the code's generating polynomial; the register is then successively shifted one bit position in the direction of lower order. The discussion covers burst correction with m-bit deletion, burst/deletion decoder implementation, false correction probability, and bit slippage involving bit insertions. The principal elements of the burst/deletion correction decoder are presented in schematic form.
A problem in designing semiconductor memories is to provide some measure of error control without requiring excessive coding overhead or decoding time. In LSI and VLSI technology, memories are often organized on a multiple bit (or byte) per chip basis. For example, some 256 K bit DRAM's are organized in 32 K x 8 bit-bytes. Byte-oriented codes such as Reed-Solomon (RS) codes can provide efficient low overhead error control for such memories. However, the standard iterative algorithm for decoding RS codes is too slow for these applications. The paper presents a special decoding technique for double-byte-error-correcting, triple-byte-error-detecting RS codes which is capable of high-speed operation. This technique is designed to find the error locations and the error values directly from the syndrome without having to use the iterative algorithm to find the error locator polynomial.
Bose-Chaudhuri-Hocquenghem codes for correcting random errors
A decoding method for a (23,12) Golay code is extended to the important 1/2-rate (24,12) Golay code so that three errors can be corrected and four errors can be detected. It is shown that the method can be extended to any decoding method which can correct three errors in the (23,12) Golay code.