Noiseless Coding Research Articles

Hadamard matrices as a result of Scarpis product without cyclic shifts

Introduction: Orthogonal Hadamard matrices consisting of elements 1 and –1 (real number) exist for orders that are multiples of 4. The study considers the product of an orthogonal Hadamard matrix and its core, which is called the Scarpis product, and is similar in meaning to the Kronecker product. Purpose: To show by revealing the symmetries of the block Hadamard matrices that their observance contributes to a product that generalizes the Scarpis method to the nonexistence of a finite field. Results: The study demonstrates that orthogonality is an invariant of the product under discussion, subject to the two conditions: one of the multipliers is inserted into the other one, the sign of the elements of the second multiplier taken into account (the Kronecker product), but with a selective action of the sign on the elements and, most importantly, with the cyclic permutation of the core which depends on the insertion location. The paper shows that such shifts can be completely avoided by using symmetries that are characteristic of the universal forms of Hadamard matrices. In addition, this technique is common for many varieties of adjustable Kronecker products. Practical relevance: Orthogonal sequences and effective methods for their finding by the theory of finite fields and groups are of direct practical importance for the problems of noiseless coding, video compression and visual masking.

Construction of an orthogonal basis based on pseudorandom sequences

In this paper, we consider an approach to constructing an orthogonal transform based on pseudorandom sequences with the purpose of providing noiseless coding. This approach ensures the absence of congestion in the channel thus allowing the problem of restricted access in confidential systems to be solved.

Improving Noise Immunity And Efficiency Using High-precision Iterative Codes

The article discusses the issues of ensuring noise immunity in digital broadcasting systems, shows the importance of the transition to the optimal code and the need to use it in the field of noiseless coding in various areas of telecommunication transmission and reception of digital signals. The previous algorithms and error-correcting coding methods based on the Gray code, which are used in multi-level digital broadcast modulation schemes to minimize the intensity of bit errors, are highlighted. A model of error-correcting coding by the Gray method and methods for estimating the probability of error for the Gray code are presented. Based on computer modeling in the Matlab 7.0 Simulink environment, a model of a noise-resistant coding system was developed, which works on the basis of a parallel-cascade high-precision iterative coding and decoding algorithm, a method for determining and estimating the probability of error is given for the high-precision iterative coding and decoding algorithm, and the complexity of constructing a high-precision iterative code. The study obtained probabilistic-energy characteristics for the Gray code and for a high-precision iterative code in various positions of phase manipulation. A comparative analysis of the energy gain G (dB) of the high-precision iterative coding algorithm with the Gray coding algorithm is performed. The simulation results in a Simulink environment of an error-correcting Gray code and a high-precision iterative code in a digital information transfer system are presented

Noiseless coding algorithm based on cognitive processing of biometric data in system of digital identification of passengers

The new approach to increasing the reliability and validity of the transmission of biometric information in the digital passenger identification system based on the use of noiseless coding algorithms with elements of cognitive data processing is presented in the article. Reed-Solomon non-binary codes as best suited for protecting digitized images are given to use in the proposed permutation decoding method data protection against errors. When integrating the cognitive map into the model of the permutation decoder, it is possible to implement the learning function. The training consists in that the decoder records the results of the transformations of the generating matrices of equivalent codes into the memory of a cognitive map using a ready solution under the conditions of permutations repetition. Thereby, a reduction in the volume of arithmetic calculations is achieved with use of only fast matrix transformations. The proposed method has a number of properties allowing one to reduce the e memory cognitive maps volume. The simulation results have shown that the implementation of the proposed method makes it possible to gain speeding up the calculations by reducing the number of operations.

Synthesis of a Concurrent Error Detection Circuit Based on the Spectral R-Code with the Partitioning of Outputs into Groups

The pace of development of the microelectronic industry and the progress in the development of computer-aided design tools make problems associated with the development of combinational circuits that are resistant to short self-clearing faults relevant again. These faults occur due to a combination of many different factors, such as extreme operating conditions and transition to nanometer design standards. Structural redundancy methods are often used to solve this problem by the principles of the noiseless coding theory to protect information during its transmission over communication channels. However, these methods have a significant drawback, i.e., large structural redundancy. In this paper, we propose to use an approach based on the synthesis of fault-tolerant combinational circuits based on the spectral R-code to solve the problem of developing fault tolerant combinational circuits. If a specific part of the coder is protected using special technological means, this code can correct a single-bit error and detect a double-bit error. The resulting circuit has less structural redundancy compared to the traditional method of triple modular redundancy (TMR). An approach is also proposed based on partitioning circuit outputs into groups followed by synthesizing fault-tolerant combinational circuits based on the R-code, which increases the probability of error detection/correction, to minimize the probability of multiple-bit errors within a combinational circuit. The paper presents the results of a series of numerical experiments that show the efficiency of the proposed approaches.

Credibility distribution based R-norm fuzzy information measures in noiseless coding theorems

ABSTRACTIn the present communication, a parametric R-norm fuzzy entropy of a fuzzy variable and subsequently R-norm fuzzy directed divergence measure of fuzzy variables by taking the credibility distribution measure into account have been proposed along with their proof of validity. Based on the proposed information measures, some new results along with its various properties and the monotonic behavior have been studied in detail. Further, some important results and inequalities related to their corresponding noiseless coding theorems have been well established. Some particular cases related to the proposed fuzzy information measures and fuzzy mean codeword length in terms of existing results have also been discussed. In view the monotonic behavior of the fuzzy mean codeword length, an application to chose the appropriate value of the parameter R to reduce the fuzzy mean codeword length significantly for optimal code has been presented with a possible variability.

Some Inequalities in Information Theory Using Tsallis Entropy

We presenta relation betweenTsallis’s entropy and generalizedKerridge inaccuracywhich is called generalizedShannon inequalityand is well-known generalization ininformation theoryand then give its application in coding theory. The objective of the paper is to establish a result on noiseless coding theorem for the proposed mean code length in terms of generalized information measure of orderξ.

Эффективность применения помехоустойчивого кодирования при цифровом дифференцировании сигналов

Эффективность применения помехоустойчивого кодирования при цифровом дифференцировании сигналов

Noiseless Coding Theorems of Generalized Useful Fuzzy Inaccuracy Measure of Order a and Type

In this paper, we present a new generalized useful fuzzy inaccuracy measure and generalized fuzzy code-word length of order and type . These measures are not only new but some known measures are the particular cases of our proposed measures. We have also obtained the bounds of generalized fuzzy code-word length in terms of generalized useful fuzzy inaccuracy measure.

(R, S)-Norm Information Measure and A Relation Between Coding and Questionnaire Theory

In this paper, we introduce a quantity which is called (R, S)-norm entropy and discuss some of its major properties in comparison with Shannon’s and other entropies known in the literature. Further, we give an application of (R, S)-norm entropy in coding theory and a coding theorem analogous to the ordinary coding theorem for a noiseless channel. The theorem states that the proposed entropy is the lower bound of mean code word length. Further, we give an application of (R, S)-norm entropy and noiseless coding theorem in questionnaire theory. We show that the relationship between noiseless coding theorem and questionnaire theory through a charging scheme based on the resolution of questions and lower bound on the measure of the charge can also be obtained.

Features of the Data Transmission in the Wireless Capsule Endoscopic Complex

This article describes the features of the data transmission between a wireless endoscopic capsule and a reading device. The advantages and disadvantages of using a uni-and a bidirectional communication in the capsule are considered and the unidirectional communication is chosen as a preferred one. The various embodiments of the integrity control and the recovery are analyzed and experimentally verified. To justify the choice between the noiseless coding and the checksums we carried out an experiment on the effect of their implementation on energy consumption of the endoscopic capsule. It was found that the noiseless coding significantly reduced the battery life of the capsule and the choice was made in favor of the checksums.

Maximal noiseless code rates for collective rotation channels on qudits

We study noiseless subsystems on collective rotation channels of qudits, i.e., quantum channels with operators in the set ${\mathcal E}(d,n) = \{ U^{\otimes n}: U \in {\mathrm{SU}}(d)\}.$ This is done by analyzing the decomposition of the algebra ${\mathcal A}(d,n)$ generated by ${\mathcal E}(d,n)$. We summarize the results for the channels on qubits ($d=2$), and obtain the maximum dimension of the noiseless subsystem that can be used as the quantum error correction code for the channel. Then we extend our results to general $d$. In particular, it is shown that the code rate, i.e., the number of protected qudits over the number of physical qudits, always approaches 1 for a suitable noiseless subsystem. Moreover, one can determine the maximum dimension of the noiseless subsystem by solving a non-trivial discrete optimization problem. The maximum dimension of the noiseless subsystem for $d = 3$ (qutrits) is explicitly determined by a combination of mathematical analysis and the symbolic software Mathematica.

Comparative Study of Generalized Quantitative-Qualitative Inaccuracy Fuzzy Measures for Noiseless Coding Theorem and 1:1 Codes

In coding theory, we study various properties of codes for application in data compression, cryptography, error correction, and network coding. The study of codes is introduced in Information Theory, electrical engineering, mathematics, and computer sciences for the transmission of data through reliable and efficient methods. We have to consider how coding of messages can be done efficiently so that the maximum number of messages can be sent over a noiseless channel in a given time. Thus, the minimum value of mean codeword length subject to a given constraint on codeword lengths has to be founded. In this paper, we have introduced mean codeword length of orderαand typeβfor 1:1 codes and analyzed the relationship between average codeword length and fuzzy information measures for binary 1:1 codes. Further, noiseless coding theorem associated with fuzzy information measure has been established.

A Noiseless Coding Theorem Connected with Generalized Renyi’s Entropy of Order α for Incomplete Power Probability Distribution pβ

A Noiseless Coding Theorem Connected with Generalized Renyi’s Entropy of Order α for Incomplete Power Probability Distribution pβ

Some More Noiseless Coding Theorem on Generalized R-Norm Entropy

A parametric mean length is defined as the quantity$L_{R}^{\beta}=\frac{R}{R-1}\left[1-\frac{\sum_{i=1}^{N}p_{i}^{\beta}D^{-n_{i}\left(\frac{R-1}{R}\right)}}{\sum_{j=1}^{N}p_{j}^{\beta}}\right]$\noindent where $R>0\left(\ne 1\right),\beta >0,\, p_{i} >0,\, \sum p_{i} =1,\, \, i=1,2,\dots,N.$ This beingthe mean length of code words. Lower and upper bounds for $L_{R}^{\beta}$ are derived in terms of R-norminformation measure for the incomplete power distribution.

Adiabatic Markovian Dynamics

We propose a theory of adiabaticity in quantum markovian dynamics based on a decomposition of the Hilbert space induced by the asymptotic behavior of the Lindblad semigroup. A central idea of our approach is that the natural generalization of the concept of eigenspace of the Hamiltonian in the case of markovian dynamics is a noiseless subsystem with a minimal noisy cofactor. Unlike previous attempts to define adiabaticity for open systems, our approach deals exclusively with physical entities and provides a simple, intuitive picture at the Hilbert-space level, linking the notion of adiabaticity to the theory of noiseless subsystems. As two applications of our theory, we propose a general framework for decoherence-assisted computation in noiseless codes and a dissipation-driven approach to holonomic computation based on adiabatic dragging of subsystems that is generally not achievable by nondissipative means.

The Effect of System Load on the Existence of Bit Errors in CDMA With and Without Parallel Interference Cancelation

In this correspondence, we study a lightly loaded code-division multiple-access (CDMA) system with and without multistage hard- and soft-decision parallel interference cancelation (HD-PIC and SD-PIC). Throughout this paper we will only consider the situation of a noiseless channel, equal powers and random spreading codes. For the system with no or a fixed number of steps of interference cancelation, we give a lower bound on the maximum number of users such that the probability for the system to have no bit-errors converges to one. Moreover, we investigate when the matched filter system, where parallel interference cancelation is absent, has bit errors with probability converging to one. This implies that the use of HD-PIC and SD-PIC significantly enhances the number of users the system can serve

Partial quantum information

Information--be it classical or quantum--is measured by the amount of communication needed to convey it. In the classical case, if the receiver has some prior information about the messages being conveyed, less communication is needed. Here we explore the concept of prior quantum information: given an unknown quantum state distributed over two systems, we determine how much quantum communication is needed to transfer the full state to one system. This communication measures the partial information one system needs, conditioned on its prior information. We find that it is given by the conditional entropy--a quantity that was known previously, but lacked an operational meaning. In the classical case, partial information must always be positive, but we find that in the quantum world this physical quantity can be negative. If the partial information is positive, its sender needs to communicate this number of quantum bits to the receiver; if it is negative, then sender and receiver instead gain the corresponding potential for future quantum communication. We introduce a protocol that we term 'quantum state merging' which optimally transfers partial information. We show how it enables a systematic understanding of quantum network theory, and discuss several important applications including distributed compression, noiseless coding with side information, multiple access channels and assisted entanglement distillation.

Universal codes for finite sequences of integers drawn from a monotone distribution

We offer two noiseless codes for blocks of integers X/sup n/ = (X/sub 1/, ..., X/sub n/). We provide explicit bounds on the relative redundancy that are valid for any distribution F in the class of memoryless sources with a possibly infinite alphabet whose marginal distribution is monotone. Specifically, we show that the expected code length L (X/sup n/) of our first universal code is dominated by a linear function of the entropy of X/sup n/. Further, we present a second universal code that is efficient in that its length is bounded by nH/sub F/ + o(nH/sub F/), where H/sub F/ is the entropy of F which is allowed to vary with n. Since these bounds hold for any n and any monotone F we are able to show that our codes are strongly minimax with respect to relative redundancy (as defined by Elias (1975)). Our proofs make use of the elegant inequality due to Aaron Wyner (1972).

Thermodynamic Interpretation of the Quantum Error Correcting Criterion

Shannon's fundamental coding theorems relate classical information theory to thermodynamics. More recent theoretical work has been successful in relating quantum information theory to thermodynamics. For example, Schumacher proved a quantum version of Shannon's 1948 classical noiseless coding theorem. In this note, we extend the connection between quantum information theory and thermodynamics to include quantum error correction. There is a standard mechanism for describing errors that may occur during the transmission, storage, and manipulation of quantum information. One can formulate a criterion of necessary and sufficient conditions for the errors to be detectable and correctable. We show that this criterion has a thermodynamical interpretation. PACS: 03.67; 05.30; 63.10