Probability Of Undetected Error Research Articles

Low-density and high-density asymmetric CT-burst correcting integer codes

<p style='text-indent:20px;'>Two classes of integer codes correcting low-density and high-density asymmetric CT-bursts within a <inline-formula><tex-math id="M1">\begin{document}$ b $\end{document}</tex-math></inline-formula>-bit byte have been presented here. Unlike the previously studied integer codes correcting burst errors, here we study such codes with weight constraint on the burst. The proposed codes are compared with similar integer codes in terms of various properties, viz. memory consumption and number of table look ups. This article winds up with an idea of determining the probability of erroneous decoding and an approach for undetected error probability.</p>

Bounds on the Probability of Undetected Error for q-Ary Codes.

We study the probability of an undetected error for general q-ary codes. We give upper and lower bounds on this quantity, by the Linear Programming and the Polynomial methods, as a function of the length, size, and minimum distance. Sharper bounds are obtained in the important special case of binary Hamming codes. Finally, several examples are given to illustrate the results of this paper.

Enhancement of safety communication model

Abstract A fundamental measure of safety communication is the residual error probability, i. e., the probability of undetected errors. For the detection of data errors, typically a Cyclic Redundancy Check (CRC) is applied, and the resulting residual error probability is determined based on the Binary Symmetric Channel (BSC) model. The use of this model had been questioned since several error types cannot be sufficiently described. Especially the increasing introduction of security algorithms into underlying communication layers requires a more adequate channel model. This paper introduces an enhanced model that extends the list of considered data error types by combining the BSC model with a Uniformly Distributed Segments (UDS) model. Although models beyond BSC are applied, the hitherto method of the calculation of the residual error probability can be maintained.

MLED: A Layered Architecture for Reducing Undetected Error Probability in File Transfer

Reliability in petabyte-scale file transfers is critical for data collected from scientific instruments. We introduce a Multi-Layer Error Detection (MLED) architecture that significantly reduces the Undetected Error Probability (UEP) in file transfer. MLED is parameterized by a number of layers $n$ , and a policy $P_{i}$ for each layer $1 \le i \le n$ that describes its operation. MLED generalizes existing error detection approaches used in file transfer. We show conditions under which MLED reduces UEP. Analytical results show that a petabyte-size file transfer in MLED with $n=2$ using CRC-32s improves UEP by $2.49E\,+\,28$ compared to a single-layer CRC-64, when the BER is 10−10.

A Moment Ratio Bound for Polynomials and Some Extremal Properties of Krawchouk Polynomials and Hamming Spheres

Let p ≥ 2. We improve the bound ||f|| <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> / ||f|| <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ≤ (p-1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s/2</sup> for a polynomial f of degree s on the boolean cube {0,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , which comes from hypercontractivity, replacing the right hand side of this inequality by an explicit bivariate function of p and s, which is smaller than (p-1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s/2</sup> for any p > 2 and s > 0. We show the new bound to be tight, within a smaller order factor, for the Krawchouk polynomial of degree s. This implies several nearly-extremal properties of Krawchouk polynomials and Hamming spheres (equivalently, Hamming balls). In particular, Krawchouk polynomials have (almost) the heaviest tails among all polynomials of the same degree and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> The Hamming spheres have the following approximate edge-isoperimetric property: For all 1 ≤ s ≤ n/2, and for all even distances 0 ≤ i ≤ 2s(n-s) / n, the Hamming sphere of radius s contains, up to a multiplicative factor of O(i), as many pairs of points at distance i as possible, among sets of the same size (there is a similar, but slightly weaker and somewhat more complicated claim for all distances). This also implies that Hamming spheres are (almost) stablest with respect to noise among sets of the same size. In coding theory terms this means that a Hamming sphere (equivalently a Hamming ball) has the maximal probability of undetected error, among all binary codes of the same rate. We also describe a family of hypercontractive inequalities for functions on {0,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , which improve on the `usual' “ q → 2” inequality by taking into account the concentration of a function (expressed as the ratio between its <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> norms), and which are nearly tight for characteristic functions of Hamming spheres. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> This has to be interpreted with some care.

ОЦІНКА ВЛАСТИВОСТЕЙ СИНТЕЗОВАНИХ НА ОСНОВІ ТЕОРІЇ РЕШІТОК СИГНАЛЬНО-КОДОВИХ КОНСТРУКЦІЙ ДЛЯ НЕРОЗДІЛЬНИХ ФАКТОРІАЛЬНИХ КОДІВ

One of the main functions of computer systems and networks is to ensure the security of information circulating in the system. At the same time, an equally important function is to protect information from the effects of interference in the communication channel. The use of factorial codes is one approach to combining noise-tolerant encryption and cryptographic protection. But these codes are vulnerable to even errors, which lead to the transformation of one permutation from the allowed set to another permutation belonging to the same set. Therefore, the issue of improving the reliability of data transmission in systems with factorial coding is relevant. The purpose of the study is to evaluate the method of forming a signal-code structure, which ensures the achievement of the required value of transmission reliability and allows to maximize the code speed by forming a subset of permutations based on lattice theory. The relative transmission rate and the probability of undetected error as a result of the use of synthesized signal-code constructs in systems with integral factorial coding are subject to evaluation. To evaluate the efficiency of the synthesized signal-code constructions, the methods of factorial coding with data recovery by permutation and factorial coding with data recovery by permutation with error correction have been used. In order to determine the probability of undetected error, a software model has been developed that simulates a data transmission environment with independent bit errors. The results of the software model have made it possible to establish the dependence of the probability of undetected error in systems with integral factorial coding on the probability of bit error in channels with independent bit errors. A comparative analysis of the relative transmission rate and the probability of undetected error in systems with factorial coding with data recovery by permutation and factorial coding with data recovery by permutation with error correction is also performed. The results obtained in this paper have made it possible to determine the main advantages and disadvantages of the method of forming signal-code structures based on the lattice theory, as well as to determine the scope of this method.

Landmark Augmentation for Mobile Robot Localization Safety

As more robots, such as autonomous vehicles, are deployed in life-critical situations it is imperative to consider safety, and in particular, localization safety. While it would be ideal to guarantee localization safety without having to modify the environment, this is not always possible and one may have to add landmarks or active beacons for landmark-based localization. As such, this work introduces a method to identify the minimum places in an environment where landmarks can be added to ensure localization safety, as quantified using integrity risk, the probability of undetected sensor errors causing localization failure while accounting for measurement faults. The letter formulates the problem as a systematic minimization: given the robot's trajectory and the current landmark map, add the minimum number of new landmarks such that the integrity risk along the trajectory is below a given safety threshold. The letter proposes three algorithms: a naive approach, Integrity-based Landmark Generator (I-LaG), and Fast I-LaG. The computationally expensive naive algorithm serves as a reference to illustrate simple scenarios. I-LaG adds relatively fewer landmarks than the Fast I-Lag algorithm but is more computationally expensive. Simulation and experimental results validate the proposed algorithms.

A long period quasi-random sequences false synchronization probability estimation under organized interference conditions

It is known that the broadband communication systems functioning always begins with a long period quasi-random sequences sensors synchronization. In a false synchronization case, which is immediately detected by the received signal convolution absence, resynchronization can be carried out in organized interference difficult conditions from the intruders’ side. By now, a false start probability estimating issues remain open due to the obtaining mathematical expressions’ difficulty without knowing the errors’ distribution laws that lead to false start of a long period quasi-random sequences sensors synchronization. In this paper, theorems are proved on a long period quasi-random sequences sensors synchronization process isomorphism by the Ward method to the check relations fulfilment (k+m, k) of the maximum length code, which makes it possible using the coding theory theoretical apparatus to obtain fairly simple mathematical expressions for calculating a long period quasi-random sequences sensors synchronization false synchronization probability through the maximum length code undetected error (k+m, k) probability.

Distance increasing mapping for variable distance block code

A new idea regarding the generation of a block code for a code rate of 1/2 and less than 1/2 is developed here. The minimum distance between the codewords can be maximised using a suitable transformation of the P matrix in the generator matrix G of the block code. This study gives the construction of distance increasing mapping (DIM) such that the proposed code is a variable distance modified block code with a code rate equal to 1/2. An algorithm for such DIM is developed here. The construction for distance preserving mapping (DPM) is also generalised in an algorithmic way. Further, to achieve DIM from the proposed DPM, for the established codes with a code rate less than 1/2, transformation is developed with the help of permutation arrays. The performance of the codes has also been evaluated. Decoding of this code is also simple and can be carried out using the existing techniques of decoding of block codes. The probability of undetected error is found to be better than the existing block codes.

Hard decision of the zero a posteriori LLR in 5G NR

The a posteriori log likelihood ratios (LLRs) have three kinds of values: negative, zero and positive. For the binary code, the hard decision is 1 when the a posteriori LLR is negative and is 0 when the a posteriori LLR is positive. To simplify the implementation of the decoder, the hard decision is usually 0 when the a posteriori LLR is zero since sign bits of zero and positive are the same. In 5G NR, this scheme increases the undetected error probability and the hard decision should be 1 when the a posteriori LLR is zero.

On Lower Bounds on the Spectrum of a Binary Code

We refine a lower bound on the spectrum of a binary code. We give a simple derivation of the known bound on the undetected error probability of a binary code.

Reliable Transmission of Short Packets Through Queues and Noisy Channels Under Latency and Peak-Age Violation Guarantees

This paper investigates the probability that the delay and the peak-age of information exceed a desired threshold in a point-to-point communication system with short information packets. The packets are generated according to a stationary memoryless Bernoulli process, placed in a single-server queue and then transmitted over a wireless channel. A variable-length stop-feedback coding scheme—a general strategy that encompasses simple automatic repetition request (ARQ) and more sophisticated hybrid ARQ techniques as special cases—is used by the transmitter to convey the information packets to the receiver. By leveraging finite-blocklength results, the delay violation and the peak-age violation probabilities are characterized without resorting to approximations based on larg-deviation theory as in previous literature. Numerical results illuminate the dependence of delay and peak-age violation probability on system parameters such as the frame size and the undetected error probability, and on the chosen packet-management policy. The guidelines provided by our analysis are particularly useful for the design of low-latency ultra-reliable communication systems.

An Information-Theoretic Approach to the Chipless RFID Tag Identification

In this paper, we focus on the chipless radio-frequency identification (RFID), where the tag information bits are encoded by the peak/notch pattern appeared in the frequency spectrum of the radar cross section (RCS) of the tag. In particular, we restrict our attention to a simple yet prevalent “binary” coding method, where a bit 0 or bit 1 is encoded by the absence or presence of the peak/notch, respectively. We provide an information-theoretic framework for the tag identification based on such a binary coding method. Our aim is to accommodate more bits in the limited bandwidth without degrading the identification performance. To this end, we first formulate the detection of each bit as a binary asymmetric channel (where a signal processing approach is integrated into each interrogation to enhance the underlying channel quality). Moreover, it is proposed to perform multiple interrogations with majority rule-based detection (in correspondence to the signal processing approach in each interrogation). Furthermore, we introduce some error-detecting codes to further improve the performance of tag identification. For instance, motivated by the asymmetric property of the channel model, we propose to apply the constant weight code and the Berger–Freiman code (as two representatives of non-systematic and systematic codes, respectively) to the problem to be addressed in this paper. In addition, an investigation is also conducted into the cyclic redundancy check (CRC) codes (as a representative of those codes that are not dedicated to the binary asymmetric channel but could be potentially competitive for error detection). The system’s performance is analyzed through the key parameters, namely the successful transmission rate, the false identification rate (i.e., the probability of undetected errors), and the expected number of retransmissions/interrogations. The effectiveness of the proposed methods is demonstrated by the numerical results.

On the property of properness in the extended triple-error-correcting BCH codes

The undetected error probability Pu (€) for the primitive triple-error-correcting BCH codes of blocklength 2m - 1 on a BSC with cross-over probability € ≤ 1/2 has been widely studied. In this correspondence, extended triple-error-correcting BCH codes of blocklength 2m are studied and the results presented in are supported and strengthened but with a different approach i.e., it is proved that the extended triple-error-correcting BCH codes of blocklength 2m don’t satisfy the properties of proper codes for even m ≥ 6, m integer.

Streaming Data Transmission in the Moderate Deviations and Central Limit Regimes

We consider streaming data transmission over a discrete memoryless channel. A new message is given to the encoder at the beginning of each block and the decoder decodes each message sequentially, after a delay of $T$ blocks. In this streaming setup, we study the fundamental interplay between the rate and error probability in the central limit and moderate deviations regimes and show that: 1) in the moderate deviations regime, the moderate deviations constant improves over the block coding or non-streaming setup by a factor of $T$ and 2) in the central limit regime, the second-order coding rate improves by a factor of approximately $\sqrt {T}$ for a wide range of channel parameters. For both the regimes, we propose coding techniques that incorporate a joint encoding of fresh and previous messages. In particular, for the central limit regime, we propose a coding technique with truncated memory to ensure that a summation of constants, which arises as a result of applications of the central limit theorem, does not diverge in the error analysis. Furthermore, we explore interesting variants of the basic streaming setup in the moderate deviations regime. We first consider a scenario with an erasure option at the decoder, i.e., the decoder can output an erasure symbol instead of a message estimate, and show that both the exponents of the total error and the undetected error probabilities improve by factors of $T$ . Next, by utilizing the erasure option, we show that the exponent of the total error probability can be improved to that of the undetected error probability (in the order sense) at the expense of a variable decoding delay.

MODELLING OF SAFETY-RELATED COMMUNICATIONS FOR RAILWAY APPLICATIONS

Purpose. The authors according to requirements of the standard valid for safety-related communication between interlocking systems (EN 50159) analyze the situation of message undetected which is transmitted across BSC channel. The main part is orientated to description of model with CRC code used for messages assurance across noise communication channel. Methodology. For mathematical description of encoding, decoding, error detection and error correction is used algebra of polynomials. Findings. The determination of the intensity of dangerous failure caused by the electromagnetic interference was calculated. Originality. To obtain information on the probability of undetected error in transmission code and safety code and on the intensity of dangerous failure from the motel it was created program with graphical interface. To calculate the probability of undetected error for any block code (n, k) was created a supporting program that displays the probabilityof undetected error for selected interval of error bit rate. Practical value.From the measured and calculated values obtained by the simulation one can see that with increasing error bit rate is increasing also intensity of dangerous failures. Transmission code did not detect all corrupted messages therefore it is necessary to use safety code independent on transmission code in safety-related applications. CRC is not able to detect errors if all bits are logical 0.

Asymmetric Evaluations of Erasure and Undetected Error Probabilities

The problem of channel coding with the erasure option is revisited for discrete memoryless channels. The interplay between the code rate, the undetected and total error probabilities is characterized. Using the information spectrum method, a sequence of codes of increasing blocklengths $n$ is designed to illustrate this tradeoff. Furthermore, for additive discrete memoryless channels with uniform input distribution, we establish that our analysis is tight with respect to the ensemble average. This is done by analyzing the ensemble performance in terms of a tradeoff between the code rate, the undetected and total errors. This tradeoff is parameterized by the threshold in a generalized likelihood ratio test. Two asymptotic regimes are studied. First, the code rate tends to the capacity of the channel at a rate slower than $n^{-1/2}$ corresponding to the moderate deviations regime. In this case, both error probabilities decay subexponentially and asymmetrically. The precise decay rates are characterized. Second, the code rate tends to capacity at a rate of $n^{-1/2}$ . In this case, the total error probability is asymptotically a positive constant, while the undetected error probability decays as $\exp (- bn^{ 1/2})$ for some $b>0$ . The proof techniques involve the applications of a modified (or shifted) version of the Gartner–Ellis theorem and the type class enumerator method to characterize the asymptotic behavior of a sequence of cumulant generating functions.

Convolutional-Code-Specific CRC Code Design

Cyclic redundancy check (CRC) codes check if a codeword is correctly received. This paper presents an algorithm to design CRC codes that are optimized for the code-specific error behavior of a specified feedforward convolutional code. The algorithm utilizes two distinct approaches to computing undetected error probability of a CRC code used with a specific convolutional code. The first approach enumerates the error patterns of the convolutional code and tests if each of them is detectable. The second approach reduces complexity significantly by exploiting the equivalence of the undetected error probability to the frame error rate of an equivalent catastrophic convolutional code. The error events of the equivalent convolutional code are exactly the undetectable errors for the original concatenation of CRC and convolutional codes. This simplifies the computation because error patterns do not need to be individually checked for detectability. As an example, we optimize CRC codes for a commonly used 64-state convolutional code for information length k=1024 demonstrating significant reduction in undetected error probability compared to the existing CRC codes with the same degrees. For a fixed target undetected error probability, the optimized CRC codes typically require 2 fewer bits.

Detection Analysis of CRC-Assisted Decoding

Cyclic redundancy check (CRC) codes are used for error detection in many communication systems. Attempts to take advantage of the CRC bits to also improve the error correction performance have been made. In this work, the concurrent use of CRC codes in both error detection and error correction is investigated. The Hamming weight enumerator of the LTE CRC code is found to analyze its undetected error probability (UEP) and its false alarm probability (FAP). Theoretical bounds on the UEP and FAP with concurrent error detection and correction are derived. For theoretical analysis, a hypothetical sphere decoder is assumed, from which insights and recommendations for practical implementations are made.

Undetected Error Probability for Quantum Codes

From last fourteen years the work on undetected error probability for quantum codes has been silent. The undetected error probability has been discussed by Ashikhmin [3] in which it was proved that the average probability of undetected error for a given code is essentially given by a function of its weight enumerators. In this paper, new upper bounds on undetected error probability for quantum codes used for error detection on depolarization channel are given. It has also been established that the probability of undetected errors for quantum codes over depolarization channel do satisfy the upper bound analogous to classical codes.