Algebraic Codes Research Articles

Conic and cyclidic sections in double conformal geometric algebra G8,2 with computing and visualization using Gaalop

The geometric algebra, also called the double conformal/Darboux cyclide geometric algebra (DCGA), has entities that represent conic sections. DCGA also has entities that represent planar sections of Darboux cyclides, which are called cyclidic sections in this paper. This paper presents these entities and many operations on them, in terms of algebraic expressions, computer code, and many examples of computer‐generated graphics. Operations include reflection, projection, rejection, and intersection with respect to spheres and planes. Other operations include rotation, translation, and dilation. Possible applications are introduced, which include orthographic and perspective projections of conic sections onto view planes, which may be of interest in computer graphics or other computational geometry subjects

Algebraic structure of cyclic and negacyclic codes over a finite chain ring alphabet and applications

Foundational and theoretical aspects of algebraic coding theory are discussed with the concentration in the classes of cyclic and negacyclic codes over finite chain rings. The significant role of finite rings as alphabets in coding theory is presented. We surveys results on both simple-root and repeated-root cases of such codes. Many directions in which the notions of cyclicity and negacyclicity have been generalized are also considered. The paper is devoted to giving an introduction to this area of applied algebra. We do not intend to be encyclopedic, the topics included are bounded to reflect our own research interest.

A Cryptographic System Based on a New Class of Binary Error-Correcting Codes

Abstract In this paper we introduce a new cryptographic system which is based on the idea of encryption due to [McEliece, R. J. A public-key cryptosystem based on algebraic coding theory, DSN Progress Report. 44, 1978, 114–116]. We use the McEliece encryption system with a new linear error-correcting code, which was constructed in [Hannusch, C.—Lakatos, P.: Construction of self-dual binary 22 k, 22 k−1, 2 k-codes, Algebra and Discrete Math. 21 (2016), no. 1, 59–68]. We show how encryption and decryption work within this cryptosystem and we give the parameters for key generation. Further, we explain why this cryptosystem is a promising post-quantum candidate.

Semi–analytical, piecewise temperature–time distributions in solid bodies of regular shape affected by uniform surface heat flux employing the Method Of Lines (MOL) and the eigenvalue method

The central objective of the present study is to generate an approximate, semi–analytical piecewise solution of the one-dimensional, unsteady heat conduction equation in regular solid bodies (large wall, long cylinder and sphere) with uniform heat flux at the exposed surfaces. The main obstacle in these heat conduction problems lies in the boundary condition of uniform heat flux that is non-homogeneous, and of course is not conducive to the direct utilization of the method of separation of variables. To circumvent this restrictive obstacle, the traditional solution procedure relies on the principle of superposition of solutions composed of three degraded sub-solutions: one sub-solution in terms of the space coordinate and time, another sub-solution in terms of the space coordinate and another sub-solution in terms of time. In the present study, we exploit a radical different approach applying the Method Of Lines (MOL) forthwith to the one-dimensional, unsteady, heat conduction equation with prescribed uniform heat flux. Essentially, MOL discretizes the spatial partial derivatives in the one-dimensional, unsteady, heat conduction equation, while leaving the time partial derivative continuous. Correspondingly, the MOL transformation gives way to an adjoint system of linear, ordinary differential equations of first order in time, which is solved analytically (not numerically) with the potent eigenvalue method in the platform of a symbolic algebra code. The outcome of the two-part computational procedure provides a sequence of semi-analytical piecewise temperature distributions at the pre-selected lines in the large wall, long cylinder and sphere, which are expressed in terms of linear combinations of exponential functions of time comprising the required set of eigenvalues and eigenvectors. At the end, it was determined that the semi-analytical piecewise surface, center and mean temperature-time distributions based on seven lines exhibit excellent quality in the entire time domain.

Characterising bimodal collections of sets in finite groups

A collection of disjoint subsets \({\mathcal {A}}=\{A_1,A_2,\ldots ,A_m\}\) of a finite abelian group has the bimodal property if each non-zero group element \(\delta \) either never occurs as a difference between an element of \(A_i\), and an element of \(A_j\) with \(j\ne i\), or else for every element \(a_i\) in \(A_i\), there is an element \(a_j\in A_j\) for some \(j\ne i\) with \(a_i-a_j=\delta \). This property arises in familiar situations, such as cosets of a fixed subgroup or in a group partition, and has applications to the construction of optimal algebraic manipulation detection codes. In this paper, we obtain a structural characterisation for bimodal collections of sets.

A Method of Error Correction in Arithmetic Logic Devices of Information-Measuring Systems Processors

An error-correction method is proposed for computer arithmetic logic units, in which memory duplication is executed and an algebraic linear code with minimal information redundancy is used for control. Rules for the use of coded information have been formulated for error detection during execution of arithmetic and logical operations. A procedure is proposed for the formulation of the duplication channel of arithmetic logic units due to the use of equipment intended for coding of information and the functional redundancy of the arithmetic logic units of the processor. Mechanisms are revealed that make it possible to use the functional arithmetic logic units of the processor and the monitoring equipment for the execution of primary and duplicating functions. Hardware costs are calculated for the formation of the duplication channel, and a comparative estimation is conducted of the gain in probability of failure-free operation of the fail-safe computer when using the proposed method of error correction in comparison with the majority reservation method. It is shown that the developed method of error correction, by comparison with the majority method, makes it possible to improve the probability of failure-free operation of the computer, reduce by a third the overall hardware costs, and reduce by a factor of 2.5 the expenditures for creation of the arithmetic logic unit of the processor.

Computing all border bases for ideals of points

In this paper, we consider the problem of computing all possible order ideals and also sets connected to 1, and the corresponding border bases, for the vanishing ideal of a given finite set of points. In this context, two different approaches are discussed: based on the Buchberger–Möller Algorithm [H. M. Möller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, EUROCAM ’82 Conf., Computer Algebra, Marseille/France 1982, Lect. Notes Comput. Sci. 144, (1982), pp. 24–31], we first propose a new algorithm to compute all possible order ideals and the corresponding border bases for an ideal of points. The second approach involves adapting the Farr–Gao Algorithm [J. B. Farr and S. Gao, Computing Gröbner bases for vanishing ideals of finite sets of points, in 16th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC-16, Las Vegas, NV, USA (Springer, Berlin, 2006), pp. 118–127] for finding all sets connected to 1, as well as the corresponding border bases, for an ideal of points. It should be noted that our algorithms are term ordering free. Therefore, they can compute successfully all border bases for an ideal of points. Both proposed algorithms have been implemented and their efficiency is discussed via a set of benchmarks.

New Soft Set Based Class of Linear Algebraic Codes

New Soft Set Based Class of Linear Algebraic Codes

Holographic Renyi entropy from quantum error correction

We study Renyi entropies Sn in quantum error correcting codes and compare the answer to the cosmic brane prescription for computing {tilde{S}}_nequiv {n}^2{partial}_nleft(frac{n-1}{n}{S}_nright) . We find that general operator algebra codes have a similar, more general prescription. Notably, for the AdS/CFT code to match the specific cosmic brane prescription, the code must have maximal entanglement within eigenspaces of the area operator. This gives us an improved definition of the area operator, and establishes a stronger connection between the Ryu-Takayanagi area term and the edge modes in lattice gauge theory. We also propose a new interpretation of existing holographic tensor networks as area eigenstates instead of smooth geometries. This interpretation would explain why tensor networks have historically had trouble modeling the Renyi entropy spectrum of holographic CFTs, and it suggests a method to construct holographic networks with the correct spectrum.

Counting Z2 Z4 Z8-additive Codes

In Algebraic Coding Theory, all linear codes are described by generator matrices. Any linear code has many generator matrices which are equivalent. It is important to find the number of the generator matrices for constructing of these codes. In this paper, we study Z_2 Z_4 Z_8-additive codes, which are the extension of recently introduced Z_2 Z_4-additive codes. We count the number of arbitrary Z_2 Z_4 Z_8-additive codes. Then we investigate connections to Z_2 Z_4 and Z_2 Z_8-additive codes with Z_2 Z_4 Z_8, and give some illustrative examples.

Принципи програмної реалізації біоінспірованого методу декодування алгебраїчних згорткових кодів

It is shown that advisable to use algebraic convolution codes for increasing the reliability of the transmission of information in telecommunication networks based on radio technologies. An analysis of existing methods for decoding error correcting codes was carried out. It is shown that the problem of soft decoding of a given algebraic convolutional code can be formally presented as an optimization problem. The expediency of using the bioinspired approach together with the random bias procedure for soft decoding of algebraic convolution codes is substantiated. The main stages of the bioinspired decoding method of these codes are considered. For the effective software implementation of the bioinspired decoding method of these codes, the schemes of the corresponding algorithms have been developed and the features of their implementation have been determined. The received sequence, the parameters of the algebraic convolution code and the parameters of the chosen bioinspired procedure are the input data of the developed algorithm of bioinspired decoding of algebraic convolutional codes with the random bias. The output of this decoding algorithm is to estimate the most probable code sequence. The primary base vector of soft solutions, the most reliable basis of the generator matrix and the parameters of the selected bioinspired procedure are the input data in the presented algorithm of the subprocess of the bioinspired search of the codeword of the algebraic convolution codes. The output of this decoding algorithm is the best basic code sequence. Then this sequence is used in the basic algorithm for decoding algebraic convolutional codes.

Analysis of efficiency of the bioinspired method for decoding algebraic convolutional codes

It has been shown that convolutional codes are widely used, along with various decoding methods, to improve the reliability of information transmission in wireless telecommunication systems. The general principles of synthesis and the parameters and algebraic non-systematic convolutional codes with arbitrary coding rate and maximum achievable code distance have been shown. The basic stages of the bioinspired method for decoding algebraic convolutional codes using a random shift mechanism have been presented. It has been shown that the essence of the presented decoding method implies applying the procedure of differential evolution with the heuristically determined parameters. In addition, this method uses information about the reliability of the adopted symbols to find the most reliable basis for the generalized generator matrix. The mechanism of random shift for the modification of the accepted sequence is additionally applied for the bioinspired search based on various most reliable bases of a generalized generator matrix. The research results established that the bioinspired method for decoding algebraic convolutional codes ensures greater efficiency compared with the algebraic decoding method in the communication channel with additive white Gaussian noise. Depending on the parameters of the algebraic convolutional code and the necessary error coefficient, the energy gain from encoding ranges from 1.6 dB to 3 dB. It was shown that the presented bioinspired decoding method can be used for convolutional codes with a large code constraint length. In doing so, the presented method for decoding algebraic convolutional codes is less efficient than the Viterbi decoding method and turbo codes at a sufficient number of decoding iterations

Some new linear codes from skew cyclic codes and computer algebra challenges

One of the main problems of coding theory is to construct codes with best possible parameters. Cyclic codes and their various generalizations, such as quasi-twisted codes, have been a fruitful source in achieving this goal. Recently, a new generalization of cyclic codes that are known as skew cyclic codes have been introduced and some new codes obtained from this class. Unlike many other types of codes considered in algebraic coding theory, skew cyclic codes require one to work in a non-commutative ring called skew polynomial ring. In this paper, we present some new linear codes obtained from the class of skew cyclic codes and describe computational challenges in working with this class of codes.

Developing Bid-Ask Probabilities for High-Frequency Trading

Methods of path integrals are used to develop multi-factor probabilities of bid-ask variables for use in high-frequency trading (HFT). Adaptive Simulated Annealing (ASA) is used to fit the nonlinear forms so developed to a day of bitmex tick data. Maxima algebraic code is used to develop the path integral codes into C codes, and sampling code is used for the fitting process. After these fits, the resultant C code is very fast and useful for forecasting upcoming ask, bid, midprice, etc., when narrow and wide windows of incoming data are used. A bonus is the availability of canonical momenta indicators (CMI) useful to forecast direction and strengths of these variables.

Concrete Syntax with Black Box Parsers

Context: Meta programming consists for a large part of matching, analyzing, and transforming syntax trees. Many meta programming systems process abstract syntax trees, but this requires intimate knowledge of the structure of the data type describing the abstract syntax. As a result, meta programming is error-prone, and meta programs are not resilient to evolution of the structure of such ASTs, requiring invasive, fault-prone change to these programs. Inquiry: Concrete syntax patterns alleviate this problem by allowing the meta programmer to match and create syntax trees using the actual syntax of the object language. Systems supporting concrete syntax patterns, however, require a concrete grammar of the object language in their own formalism. Creating such grammars is a costly and error-prone process, especially for realistic languages such as Java and C++. Approach: In this paper we present Concretely, a technique to extend meta programming systems with pluggable concrete syntax patterns, based on external, black box parsers. We illustrate Concretely in the context of Rascal, an open-source meta programming system and language workbench, and show how to reuse existing parsers for Java, JavaScript, and C++. Furthermore, we propose Tympanic, a DSL to declaratively map external AST structures to Rascal's internal data structures. Tympanic allows implementors of Concretely to solve the impedance mismatch between object-oriented class hierarchies in Java and Rascal's algebraic data types. Both the algebraic data type and AST marshalling code is automatically generated. Knowledge: The conceptual architecture of Concretely and Tympanic supports the reuse of pre-existing, external parsers, and their AST representation in meta programming systems that feature concrete syntax patterns for matching and constructing syntax trees. As such this opens up concrete syntax pattern matching for a host of realistic languages for which writing a grammar from scratch is time consuming and error-prone, but for which industry-strength parsers exist in the wild. Grounding: We evaluate Concretely in terms of source lines of code (SLOC), relative to the size of the AST data type and marshalling code. We show that for real programming languages such as C++ and Java, adding support for concrete syntax patterns takes an effort only in the order of dozens of SLOC. Similarly, we evaluate Tympanic in terms of SLOC, showing an order of magnitude of reduction in SLOC compared to manual implementation of the AST data types and marshalling code. Importance: Meta programming has applications in reverse engineering, reengineering, source code analysis, static analysis, software renovation, domain-specific language engineering, and many others. Processing of syntax trees is central to all of these tasks. Concrete syntax patterns improve the practice of constructing meta programs. The combination of Concretely and Tympanic has the potential to make concrete syntax patterns available with very little effort, thereby improving and promoting the application of meta programming in the general software engineering context.

Algebraic List-Decoding in Projective Space: Decoding With Multiplicities and Rank-Metric Codes

The problem of list decoding algebraic subspace codes and rank-metric codes is considered. We develop two separate methods, via two different approaches, for list decoding subspace codes and rank-metric codes. These methods provide, for certain code parameters, improved tradeoffs between rate and error-correction capability than that of the Koetter-Kschischang codes, in the domain of subspace codes, and than that of the Gabidulin codes, in the domain of rank-metric codes, and several other extensions thereof. In the first approach, we introduce the notion of root multiplicities for a certain sub-ring of the ring of linearized polynomials. In the list-decoding algorithm, multiple roots are enforced for the interpolation polynomial in order to achieve an improved error-correction radius for an extended family of Koetter-Kschischang subspace codes. The normalized error-correction radius for this approach is τ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> =2(L+1)/(r+1)-1-L(L+1)(L+n)/r(r+1)R, where L is the maximum list size, n is the subspace code dimension, R is the rate of the code, and r is the multiplicity parameter. In the second approach, we construct a folded version of Koetter-Kschischang codes. A linear-algebraic list-decoding algorithm is proposed for these codes that achieves the error-correction radius τ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> =s(1-sR), where s is the folding parameter. As opposed to the first approach, the size of output list in the second approach depends on the underlying field size and is at most q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m(s-1)</sup> , where q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> is the size of the field that message symbols are chosen from. It is also shown that the output list size is 1, with high probability, in a probabilistic setting. We utilize the techniques of the second approach in the domain of rank-metric codes to construct folded Gabidulin codes to enable a linear-algebraic list-decoding algorithm for such codes. Our algorithm makes it possible to recover.

Classical and Quantum Convolutional Codes Derived From Algebraic Geometry Codes

In this paper, we construct new families of convolutional codes. Such codes are obtained by means of algebraic geometry codes. Additionally, more families of convolutional codes are constructed by means of puncturing, extending, expanding and by the direct product code construction applied to algebraic geometry codes. The parameters of the new convolutional codes are better than or comparable to the ones available in literature. In particular, a family of almost near MDS codes is presented.

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An important part of the combat activity of units of the National Guard of Ukraine is the physical protection of important state critical infrastructure facilities. The development of modern information technologies leads to the emergence of new threats to the security of such facilities, in particular, the number and power of cyber-attacks motivated by the interests of individual states, organizations and groups of people is growing. This determines the relevance of ensuring the confidentiality and integrity of messages exchanged between units of the National Guard of Ukraine through a warning system with potentially dangerous objects. To solve this problem, imitation-resistant data encoding is used, which makes it possible to check whether the data has been changed by a third party. The likelihood that the data has been changed serves as a measure of the resistance of the cipher.Ensuring the simulated resistance of data transmission systems is possible on the basis of solving an interconnected set of information protection tasks. The quality of solving security problems is largely determined by cryptographic algorithms for encryption, digital signature, hashing and generation of authentication codes used.A study of ways to increase the spoofing resistance of warning signals of the guard systems of the National Guard of Ukraine at critical infrastructure facilities was conducted.An analysis of the authenticity of messages with constructive elements of MAC codes based on hash function families is made. It is concluded that practical hashing schemes should include hash classes with a large compression ratio for data of very large volumes as possible. For these purposes, hash families on the basis of long algebraic geometric codes are of interest.An analysis of the general characteristics of algebraic-geometric codes showed that the best properties in terms of their code distance and computational complexity are the extended Reed-Solomon code, codes on Hermite curves, and codes on Suzuki curves. Collision stability estimates are presented for universal hashing schemes for the best algebraic curves with a large number of points and a maximum curve.The presented theory of universal hashing with Reed-Solomon algebraic-geometrical codes and codes on Hermite curves shows the ability to provide the necessary measure of the probability of collision of hash functions. Since a linear increase in collision probability for Reed-Solomon hashing limits the size of the hashed message, one of the ways to increase the simplicity of data transmission is to use codes on Hermite curves.

Design of MIMO-Visible Light Communication Transceiver Using Maximum Rank Distance Codes

This paper is aimed at comparing the performance of multiple-input multiple-output (MIMO) schemes applied to the indoor visible light communication (VLC) system. The MIMO schemes considered for this paper are repetition coding (RC), space-time block codes (STBCs), and spatial multiplexing (SMP). For MIMO design, the current system makes use of unique algebraic codes, i.e., maximum rank distance (MRD) codes in this connection. Two different configurations (2 × 1 and 2 × 2 ) are taken into consideration with distinct transmitters' spacing. Simulation results are presented and a comparative analysis of the current system with an existing system is given mainly in terms of bit error rate (BER), data-rate, and transmission range. The simulation results are validated by the physical implementation of the system using NI cDAQ module programmed in LabVIEW. The simulation and experimentation results indicate that the RC scheme with significant diversity gain provides more robustness as compared to other MIMO schemes; however, the RC exhibits poor multiplexing gain, and hence, it is not spectral efficient. As its counterpart, the STBC and SMP are used that can increase both the capacity and reliability at the cost of slightly reduced transmission range.

Analysis and investigation of algebraic geometric codes properties

Linear block noise-proof codes constructed according to algebraic curves (algebraic geometric codes) are considered, their design properties are evaluated, algorithms of construction and decoding are studied. The energy efficiency of the transmission of discrete messages by M-ary orthogonal signals in the application of algebraic geometric codes is studied; the achievable energy gain from the use of noise-immune coding is estimated. It is shown that in discrete channels without memory it is possible to obtain a significant energy gain, which increases with the transition to long algebraic geometric codes constructed by curves with a large number of points with respect to the genus of the curve. It is established that the computational complexity of implementing algebraic geometric codes is comparable to other known noise-resistant codes, for example, Reed-Solomon codes and others. Thus, high energy efficiency in combination with acceptable computational complexity of implementation confirm the prospects of algebraic geometric codes using in modern telecommunication systems and networks to improve the noise immunity of data transmission channels.