On the probability of undetected error for the maximum distance separable codes

The performance of maximum-distance-separable codes with symbols from GF(1) when they are used for pure error detection or for simultaneous error correction and detection is investigated. These codes are good for symbol error detection. Their probability of undetected error is upper bounded by q to the -r and decreases monotonically as the symbol error probability epsilon decreases from (q-1)/q to 0, where r is the number of parity-check symbols of a code

Undetected error probability for data services in a terrestrial DAB single frequency network

DAB (Digital Audio Broadcasting) is the European successor of FM radio. Besides audio services, other services such as traffic information can be provided.\ud An important parameter for data services is the probability of non-recognized or undetected errors in the system. To derive this probability, we propose a bound for the undetected error probability in CRC codes. In addition, results from measurements of a Single Frequency Network (SFN) in Amsterdam were used, where the University of Twente conducted a DAB field trial. The proposed error bound is compared with other error bounds from literature and the results are validated by simulations. Although the proposed bound is less tight than existing bounds, it requires no additional information about the CRC code such\ud as the weight distribution. Moreover, the DAB standard has been extended last year by an Enhanced Packet Mode (EPM) which provides extra protection for data services. An undetected error probability for this mode is also derived. In a realistic user scenario of 10 million users, a 8 kbit/s EPM sub channel can be considered as a system without any undetected errors (Pud = 6 · 10−40). On\ud the other hand, in a normal data sub channel, only 110 packets with undetected errors are received on average each year in the whole system (Pud = 5 · 10−13)

A CRC usefulness assessment for adaptation layers in satellite systems

This paper assesses the real usefulness of CRCs in today's satellite network-to-link adaptation layers under the lights of enhanced error control and framing techniques, focusing on the DVB-S and DVB-S2 standards. Indeed, the outer block codes of their FEC schemes (Reed-Solomon and BCH, respectively) can provide very accurate error-detection information to the receiver in addition to their correction capabilities, at virtually no cost. This handy feature could be used to manage on a frame-by-frame basis what CRCs do locally, on the frames' contents, saving the bandwidth and processing load associated with them, and paving the way for enhanced transport of IP over DVB-S2. Mathematical and experimental results clearly show that if FEC has been properly congured for combined error correction and detection, having an uncorrected event after FEC decoding is likely to be an extremely improbable event. Under such conditions, it seems possible and attractive to optimize the way global error-control is done over satellite links by reducing the role of CRCs, or even by removing them from the overall encapsulation process

A Decoding Algorithm for LDPC Codes Over Erasure Channels with Sporadic Errors

none4An efficient decoding algorithm for low-density parity-check (LDPC) codes on erasure channels with sporadic errors (i.e., binary error-and-erasure channels with error probability much smaller than the erasure probability) is proposed and its performance analyzed. A general single-error multiple-erasure (SEME) decoding algorithm is first described, which may be in principle used with any binary linear block code. The algorithm is optimum whenever the non-erased part of the received word is affected by at most one error, and is capable of performing error detection of multiple errors. An upper bound on the average block error probability under SEME decoding is derived for the linear random code ensemble. The bound is tight and easy to implement. The algorithm is then adapted to LDPC codes, resulting in a simple modification to a previously proposed efficient maximum likelihood LDPC erasure decoder which exploits the parity-check matrix sparseness. Numerical results reveal that LDPC codes under efficient SEME decoding can closely approach the average performance of random codes.noneG. Liva; E. Paolini; B. Matuz; M. ChianiG. Liva; E. Paolini; B. Matuz; M. Chian

Automatic-repeat-request error control schemes

Error detection incorporated with automatic-repeat-request (ARQ) is widely used for error control in data communication systems. This method of error control is simple and provides high system reliability. If a properly chosen code is used for error detection, virtually error-free data transmission can be attained. Various types of ARQ and hybrid ARQ schemes, and error detection using linear block codes are surveyed

Coding for reliable satellite communications

This research project was set up to study various kinds of coding techniques for error control in satellite and space communications for NASA Goddard Space Flight Center. During the project period, researchers investigated the following areas: (1) decoding of Reed-Solomon codes in terms of dual basis; (2) concatenated and cascaded error control coding schemes for satellite and space communications; (3) use of hybrid coding schemes (error correction and detection incorporated with retransmission) to improve system reliability and throughput in satellite communications; (4) good codes for simultaneous error correction and error detection, and (5) error control techniques for ring and star networks

Quantum Computing with Very Noisy Devices

In theory, quantum computers can efficiently simulate quantum physics, factor large numbers and estimate integrals, thus solving otherwise intractable computational problems. In practice, quantum computers must operate with noisy devices called ``gates'' that tend to destroy the fragile quantum states needed for computation. The goal of fault-tolerant quantum computing is to compute accurately even when gates have a high probability of error each time they are used. Here we give evidence that accurate quantum computing is possible with error probabilities above 3% per gate, which is significantly higher than what was previously thought possible. However, the resources required for computing at such high error probabilities are excessive. Fortunately, they decrease rapidly with decreasing error probabilities. If we had quantum resources comparable to the considerable resources available in today's digital computers, we could implement non-trivial quantum computations at error probabilities as high as 1% per gate.Comment: 47 page

Accuracy threshold for concatenated error detection in one dimension

Estimates of the quantum accuracy threshold often tacitly assume that it is possible to interact arbitrary pairs of qubits in a quantum computer with a failure rate that is independent of the distance between them. None of the many physical systems that are candidates for quantum computing possess this property. Here we study the performance of a concatenated error-detection code in a system that permits only nearest-neighbor interactions in one dimension. We make use of a new message-passing scheme that maximizes the number of errors that can be reliably corrected by the code. Our numerical results indicate that arbitrarily accurate universal quantum computation is possible if the probability of failure of each elementary physical operation is below approximately 10^{-5}. This threshold is three orders of magnitude lower than the highest known.Comment: 7 pages, 4 figures, now with error bar

A concatenated coding scheme for error control

A concatenated coding scheme for error control in data communications is analyzed. The inner code is used for both error correction and detection, however the outer code is used only for error detection. A retransmission is requested if the outer code detects the presence of errors after the inner code decoding. The probability of undetected error of the above error control scheme is derived and upper bounded. Two specific exmaples are analyzed. In the first example, the inner code is a distance-4 shortened Hamming code with generator polynomial (X+1)(X(6)+X+1) = X(7)+X(6)+X(2)+1 and the outer code is a distance-4 shortened Hamming code with generator polynomial (X+1)X(15+X(14)+X(13)+X(12)+X(4)+X(3)+X(2)+X+1) = X(16)+X(12)+X(5)+1 which is the X.25 standard for packet-switched data network. This example is proposed for error control on NASA telecommand links. In the second example, the inner code is the same as that in the first example but the outer code is a shortened Reed-Solomon code with symbols from GF(2(8)) and generator polynomial (X+1)(X+alpha) where alpha is a primitive element in GF(z(8))