Permutation Decoding Research Articles

Research of cognitive data processing in radio communication systems with permutation decoding

Background. The need to use permutation decoding tools in radio communication systems is explained by the increased error correction capabilities of this method. In this case, complex matrix calculations during the search for equivalent codes according to the classical scheme of permutation decoding are replaced by a list of ready-made solutions. These solutions are calculated a priori and entered into the cognitive cards of the decoder processor, which makes the method a convenient tool in the procedure for ensuring information reliability when controlling, for example, unmanned vehicles via radio channels. In fact, matrix calculations on board are replaced by searching the list of cognitive maps for the right solution corresponding in real time to the current permutation of reliable character numerators. However, data processing in the decoder’s cognitive map requires a special description.
 Aim. The study of methods for identifying permutations of character numerators of code vectors in order to effectively transform them in a system of cognitive maps of a permutation decoder.
 Methods. The paper reveals the subtle structure of cognitive maps of productive and unproductive permutations of numerators, which allows on a regular basis to obtain an alternative solution for switching to a set of productive permutations when the receiver receives an unproductive permutation, thereby excluding the use of trial and error.
 Results. The efficiency of the permutation decoder increases due to the implementation of permutations that were originally included in a set of solutions introduced into the cognitive map of unproductive permutations.
 Conclusion. A family of microcontrollers is proposed to implement the principle of interaction of cognitive maps with a system of alternative solutions.

Partial permutation decoding and PD-sets for [formula omitted]-linear generalized Hadamard codes

It is known that Zps-linear codes, which are the Gray map image of Zps-additive codes (linear codes over Zps), are systematic and a systematic encoding has been found. This makes Zps-linear codes suitable to apply the permutation decoding method. This technique is also based on the existence of r-PD-sets, which are subsets of the permutation automorphism group of the code. In this paper, we study the permutation automorphism group of Zps-linear generalized Hadamard codes of type (n;t1,…,ts) and show how to construct r-PD-sets of size r+1, for all r up to an upper bound, in order to be able to perform a partial permutation decoding for these codes.

Group Properties of Polar Codes for Automorphism Ensemble Decoding

In this paper, we propose an analysis of the automorphism group of polar codes, with the aim of designing codes tailored for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">automorphism ensemble</i> (AE) decoding. Using a novel description of polar codes as monomial codes through negative monomials, we prove the equivalence between the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">decreasing monomial codes</i> and the universal partial order (UPO) framework for polar codes; this property is widely believed to hold true but a formal proof was missing. We further provide a rigorous mathematical connection between code word permutations and affine transformations, an important link to understand the considered automorphisms. Based on this mathematical formalisms, we analyze the algebraic properties of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">affine automorphisms group</i> of polar codes, providing a novel description of its structure. We classify automorphisms such that all automorphisms in the same class lead to the same result under permutation decoding, which gives rise to the concept of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">redundant</i> automorphisms. Mathematically this is achieved by introducing equivalence classes of affine automorphisms under AE-based decoding. For practical application, we provide an algorithm to compute representatives for the equivalence classes, such that one automorphism from each equivalence class can be selected for use in AE decoding. A numerical analysis of the error correction performance of AE decoding of polar codes, based on equivalence classes, concludes the paper.

New advances in permutation decoding of first-order Reed-Muller codes

In this paper we describe a variation of the classical permutation decoding algorithm that can be applied to any affine-invariant code with respect to certain type of information sets. In particular, we can apply it to the family of first-order Reed-Muller codes with respect to the information sets introduced in [2]. Using this algorithm we improve considerably the number of errors we can correct in comparison with the known results in this topic.

Study of permutation decoding algorithms in UAV control systems

Relevance. This paper presents relevant topic of the use of unmanned aerial vehicles (UAV) in the agroindustrial complex (AIC) to obtain timely reliable information about the state of agricultural land for the entire period. In modern infocommunication technologies and communication systems, high requirements are placed on the reliability of information transmission. In mobile data exchange systems, compositions of noise-resistant codes are used in the form of cascade designs or turbo codes, and in order to increase their efficiency, it is necessary to make the fullest use of the redundancy introduced at each stage of data processing. To solve such a problem, it is advisable to apply a decoding method based on the ordering of the character confidence indices (CCI) of the code combination.Methods. When determining the encoding-decoding systems, the Mathcad application software was used. Mathematical studies have shown that binary block codes for reaching the specified boundary can be decoded not only on the basis of ordered statistics, but also using the method of dividing the space of allowed code combinations into clusters. Based on this method, a block diagram of the algorithm of the simulation model of the operation of the block code decoder based on cluster calculation is compiled.Results. The application of the proposed method of soft decoding of systematic block codes reduces the processing time of code combinations in the decoder by reducing the dimension of the square matrix to determine the nonlinearity property of the rows of the rearranged generating matrix after ordering the columns of the original generating matrix.

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and Their Consequences

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field $\mathrm{GF}(q)$ with length $q+1$, which are closely related to the action of the projective general linear group of degree two on the projective line. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$-ary BCH codes with length $q+1$, and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of $\mathrm{PGL} (2, p^m)$-invariant codes over $\mathrm{GF} (p^h)$ is completed. As an application of this result, the $p$-ranks of all incidence structures invariant under the projective general linear group $\mathrm{ PGL }(2, p^m)$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a $3$-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms.

МОДИФИКАЦИЯ СПОСОБА ОЦЕНИВАНИЯ МЯГКИХ РЕШЕНИЙ СИМВОЛОВ И ЕГО ВЕРИФИКАЦИЯ

To implement permutation decoding (PD), the soft symbol decisions (MSDs) are required for each element of the codeword processed by the decoder. Similar solutions are required for both binary codes and non-binary code structures [1-4]. In [5], it was shown that for binary codes it is advisable to have a maximum score equal to 710. It should be noted that the author of [5], when using integer MRS for convolutional codes, did not mean the PD method, in which it is necessary to rank symbols by reliability over the length of the code vector. The paper proves that this practice of forming the MRS turns out to be inefficient for most communication channels with additive white Gaussian noise (AWGN). It is proposed to lower the value of the index of the maximum estimate of the MPC to the value of 310 or 112 . This will halve the symbol ranking time, which improves the performance of the data receiver processors. A rigorous probabilistic estimate of the proposed approach is given.

On the Efficiency of Polar-Like Decoding for Symmetric Codes

The recently introduced polar codes constitute a breakthrough in coding theory due to their capacity-achieving property. This goes hand in hand with a quasilinear construction, encoding, and successive cancellation list decoding procedures based on the Plotkin construction. The decoding algorithm can be applied with slight modifications to Reed-Muller or eBCH codes, that both achieve the capacity of erasure channels, although the list size needed for good performance grows too fast to make the decoding practical even for moderate block lengths. The key ingredient for proving the capacity-achieving property of Reed-Muller and eBCH codes is their group of symmetries. It can be plugged into the concept of Plotkin decomposition to design various permutation decoding algorithms. Although such techniques allow to outperform the straightforward polar-like decoding, the complexity stays impractical. In this paper, we show that although invariance under a large automorphism group is valuable in a theoretical sense, it also ensures that the list size needed for good performance grows exponentially. We further establish the bounds that arise if we sacrifice some of the symmetries. Although the theoretical analysis of the list decoding algorithm remains an open problem, our result provides an insight into the factors that impact the decoding complexity.

ПЕРЕСТАНОВОЧНОЕ ДЕКОДИРОВАНИЕ В МАЛОМОЩНЫХ БЕСПРОВОДНЫХ СЕНСОРНЫХ СЕТЯХ

Wireless sensor networks are currently considered for many subject areas as effective communication systems, including, for example, industrial and agricultural facilities and enterprises, monitoring of sensors of transport security systems, medicine applications, environmental and weather monitoring, etc. In such systems, the correction code (CC) is a classic approach used to increase the reliability of the data transmission channel while simultaneously reducing the required transmitted signal power, called the energy gain code. At the same time, the power of the transmitter is reduced due to additional power consumption in the receiver decoder. Stronger codes provide better performance with lower power requirements, but have more complex decoders with higher power consumption than relatively simple error control systems. If the additional power consumption at the output of the decoder exceeds the savings in transmitted power due to the use of CC, then such a code will not be energy efficient compared to a system with a non-redundant code. Various codes can be used to protect data, while the best codes for which the critical distance turns out to be large according to a number of studies are codes with a low parity check density. The paper proposes an alternative variant based on the synthesis of a permutation decoding system using the method of localization of a limited number of permutations and the organization of additional parity checks exclusively for the verification bits of the CC using the Bayesian approach to iterative transformations of integer soft solutions of symbols.

Intelligent data processing methods in sensor networks of mobile and autonomous objects

In the near term, tasks related to monitoring mobile geographically distributed objects with the aim of efficiently managing them are acquiring great importance. An example of such tasks is the monitoring the state of given number of vehicles of agricultural enterprise, when it becomes necessary to promptly transfer given amount of data with high requirements for the level of their reliability. The use of the infrastructure of cellular mobile networks for this is not suitable for a number of reasons for rural areas. The paper proposes an approach based on the use of high-speed dynamic sensor networks. This makes it possible to carry out relay transmission of data between objects, including mobile ones. However, the issues of building flexible data transfer protocols in networks of this kind, providing a significant data transfer rate and taking into account the priority of the transmitted information and its features, still remain open. At the same time, for mobile objects in conditions of non-stationarity of the parameters of communication channels, the problems of ensuring the required reliability of data are of particular importance. To solve the last problem, it is proposed to use the permutation decoding method based on the cognitive metaphor.

Development of biometric systems for passenger identification based on noise-resistant coding means

The paper deals with the issues of using the biometric technologies to establish identity of a passenger. The purpose of the article is to analyze the techniques of enhancing reliability of various biometric identification facilities by means of using error correction codes. The basic elements and the principle of the classical biometric system functioning are presented. On the basis of the International Civil Aviation Organization (ICAO) recommendations, the procedure features of pattern recognition are presented. The versions to adopt the biometric passenger authentication procedures are under consideration. The conclusion is drawn that with the centralized biometric databases the issues of confidentiality and information security exist. The problems are characterized by the possibility of biometric images compromise, which can potentially lead to the loss of their confidentiality and the impossibility of their further usage for personal identification. The passenger authentication procedure involving the simultaneous use of biometric parameters and contact-free SMART cards seems more reliable. SMART cards are used for distributed storage of biometric and other additional data, thus neutralizing the disadvantages of access to the centralized databases. It is shown that the subsequent step in the development of this domain is the application of biometric cryptography proposing "linking" encryption keys and passwords with the biometric parameters of the subject. Consideration is given to the principle of "fuzzy extractor" operation as one of the variants for the "biometrics-code" converter. Feasibility and necessity of upgrading the means of noise-resistant coding in the systems being studied are shown. The use of permutation decoding data algorithms capable of adequately corresponding to the particular problems of biometric identification is proposed. On the basis of the results of optical communication channels statistical modeling, the necessary and sufficient conditions for application of the permutation decoding tools for binary codes are determined. The problem to minimize memory amount for the permutation decoder cognitive map due to the permutation orbits allocation and usage of the generated loops combinations as pointers of reference plane is solved. The resulting algorithm for finding a unique orbit number and its corresponding reference plane by means of receiver formation of arbitrary parameters permutation from the set of permissible permutations is proposed.

Minimal PD-sets for codes associated with the graphs $$Q^m_2$$, m even

For \(m\ge 4\) even, the duals of p-ary codes, for any prime p, from adjacency matrices for the m-ary 2-cube \(Q^m_2\) are shown to have subcodes with parameters \([m^2,2m-2,m]\) for which minimal PD-sets of size \(\frac{m}{2}\) are constructed, hence attaining the full error-correction capabilities of the code, and, as such, the most efficient sets for full permutation decoding.

ЭКСПЕРИМЕНТАЛЬНАЯ ОЦЕНКА МНОЖЕСТВА НЕПРОИЗВОДИТЕЛЬНЫХ ПЕРЕСТАНОВОК В ПРОЦЕДУРЕ ПОИСКА ЭКВИВАЛЕНТНЫХ КОДОВ

The article proposes an algorithm to search for a complete set of unproductive permutations (UP) of code vector symbols to obtain a complete cognitive decoder map (CCDM) based on a PYTHON software product by example of the Bose-ChaudhuriHocquenghem (BCH) code (15, 7, 5). Data processing in control systems based on permutation decoding (PD) of binary codes is associated with a certain probability of obtaining UP symbols of accepted code vectors that do not allow obtaining an equivalent code (EC) due to the degeneracy of generating matrices corresponding to such permutations [1-5]. Attempts to estimate this parameter using the regularities of making error-correcting codes have led to results that are approximate in nature [6, 7]. To create a CCDM, it is important to know the exact result. Otherwise, the operation of the PD system becomes inadequate, since the decoder is to maintain two contradictory operating modes during its operation. On the one hand, a fast mode based on a cognitive map, including an incomplete list of UP, and, on the other hand, a training mode, which in the case of generated permutation not included in the map, significantly delays the processing of operational information due to the required time interval for assessing the degeneracy of the rearranged generating matrix of the probable EC.

ПАРАДИГМА НЕЙРОСЕТЕВОГО ДЕКОДИРОВАНИЯ НЕДВОИЧНЫХ ИЗБЫТОЧНЫХ КОДОВ

The use of noise-tolerant coding in modern communication systems remains the only means of increasing the efficient energy of such systems. This parameter tends to increase in conditions when the receiver of the communication system is able to correct errors of a large multiplicity. At the same time, the existing experience of using various methods for decoding the received data to achieve such a goal in the format of algebraic or iterative procedures does not give a noticeable effect and leads to a large time cost and an exponential increase in the complexity of implementing the decoder processor. The reason for this situation is the passive position of the receiver, which, when processing each code vector, remains a fixator of the picture that occurred in the communication channel and, in general, by compiling a system of linear equations and then solving it, tries to identify the error vector. Some exceptions are permutation decoding systems, which, by selecting and using reliable characters from the number received at the reception, simulate the operation of their transmitter and compare the received (almost error-free) result of such encoding with the received combination [1, 2]. With the growing influence of destructive factors, such methods are ineffective. A natural question arises: are modern solutions in neural network technologies capable of improving the characteristics of code vector recognition systems in order to obtain acceptable machine time costs in order to achieve an increase in the energy characteristics of communication systems.

Perm2vec: Attentive Graph Permutation Selection for Decoding of Error Correction Codes

Error correction codes are an integral part of communication applications, boosting the reliability of transmission. The optimal decoding of transmitted codewords is the maximum likelihood rule, which is NP-hard due to the curse of dimensionality. For practical realizations, sub-optimal decoding algorithms are employed; yet limited theoretical insights prevent one from exploiting the full potential of these algorithms. One such insight is the choice of permutation in permutation decoding. We present a data-driven framework for permutation selection, combining domain knowledge with machine learning concepts such as node embedding and self-attention. Significant and consistent improvements in the bit error rate are introduced for all simulated codes, over the baseline decoders. To the best of the authors' knowledge, this work is the first to leverage the benefits of the neural Transformer networks in physical layer communication systems.

Codes from incidence matrices of some regular graphs

We examine the [Formula: see text]-ary linear codes with respect to Lee metric from incidence matrix of the Lee graph with vertex set [Formula: see text] and two vertices being adjacent if their Lee distance is one. All the main parameters of the codes are obtained as [Formula: see text] if [Formula: see text] is odd and [Formula: see text] if [Formula: see text] is even. We examine also the [Formula: see text]-ary linear codes with respect to Hamming metric from incidence matrices of Desargues graph, Pappus graph, Folkman graph and the main parameters of the codes are [Formula: see text], respectively. Any transitive subgroup of automorphism groups of these graphs can be used for full permutation decoding using the corresponding codes. All the above codes can be used for full error correction by permutation decoding.

Solution of transport safety problems based on biometric systems with error-correcting coding procedures

The paper considers models and methods for solving processing biometric data at the physical and channel levels. A classification of the parameters of such systems is given and those characteristics that play a decisive role in achieving the objective function in limited time intervals are distinguished from them. The work aims to analyze the possibilities of increasing the reliability of biometric devices at the channel level by using error correction codes. The practicability and necessity of error-correcting coding mean using the permutation data decoding algorithms capable of matching biometric identification adequately. Based on the results of statistical modelling of optical communication channels, the necessary and sufficient conditions for the application of permutation decoding tools for binary codes are determined. The problem of minimizing the memory size of the cognitive map of a permutation decoder is solved by allocating the permutation orbits and using the generating combinations of loops as indexes of the reference matrices. An efficient algorithm is proposed for finding a unique orbit number and the corresponding reference matrix when the receiver generates an arbitrary permutation of characters from the set of permissible permutations.

Codes from incidence matrices of (n, 1)-arrangement graphs and (n, 2)-arrangement graphs

We examine the p-ary linear codes from incidence matrix of the (n, k)-arrangement graphs for k = 2, n –2, n – 1. All the main parameters of the codes are obtained as respectively. We examine also the p-ary linear codes from incidence matrix of graphs such as Ljubljana graph, Heawood graph and the main parameters of the codes are [168, 111, 3] p , [21, 13, 3] p respectively. Any transitive subgroup of automorphism groups of these graphs can be used for full permutation decoding using the corresponding codes. All the above codes can be used for full error correction by permutation decoding.

Cognitive decoding of redundant block codes in the system of processing and protection of biometric data of air passengers

The article proposes a unified principle of permutation decoding (PD), which applies to redundant systematic block codes. This method allows using correction capabilities of such codes, but in the classical interpretation, the method requires cumbersome matrix calculations, which does not allow using the positive properties of the error correction method. The computing process is excessively complex. Therefore, to reduce the negative effect in the PD system, it is proposed to use the cognitive principle of data processing at the channel level, which significantly reduces the decoder complexity and ensures PD application in the air passenger biometric data systems.

ФРАКТАЛЬНАЯ КЛАСТЕРИЗАЦИЯ ГРУППОВЫХ КОДОВ В СИСТЕМЕ ВЛОЖЕННЫХ ПОЛЕЙ ГАЛУА

The permutation decoding (PD) of group systematic noise-immune codes is proved to be the most efficient method in using the redundant data entered to the code as against other methods of decoding digital data [1-5]. This opens up the opportunity of solving a complex computational problem of finding an equivalent code (EC), which is used to search for the error vector. The essence of this solution is that the computational procedure of real-time search for EC for each new combination of redundant code is replaced by a preliminary process of training the decoder to put in accordance with each new permutation of characters the generating matrix of EC parameters, which are recorded in the decoder’s memory card during training. Thus, such a memory card is called a cognitive card (CC). The article estimates the memory size of the CC, when using the block code (15,7,5), and shows the possibility of implementing a permutation decoder on basis of existing integrated circuits based on proven statements. For the first time, the apparatus of fractal partitioning of augmented binary Galois fields using the clustering of the common space of code vectors of a given code is used to prove the main statements. An efficient algorithm is presented to search for a set of invertible matrices of rearranged codes that do not provide an EC and for this reason should be primarily detected in the decoding procedure of the received code vectors.