Non-binary Codes Research Articles

Non-binary codes approach on the performance of short-packet full-duplex transmissions

This paper illustrates the enhancement of the performance of short-packet full-duplex (FD) transmission by taking the approach of non-binary low density parity check (NB-LDPC) codes over higher Galois field. For the purpose of reducing the impacts of self-interference (SI), high order of modulation, complexity, and latency decoder, a blind feedback process composed of channels estimation and decoding algorithm is implemented. In particular, this method uses an iterative process to simultaneously suppress SI component of FD transmission, estimate intended channel, and decode messages. The results indicate that the proposed technique provides a better solution than both the NB-LDPC without feedback and the binary LDPC feedback algorithms. Indeed, it can significantly improve the performance of overall system in two important factors, which are bit-error-rate (BER) and <a name="_Hlk155778233"></a>mean square error (MSE), especially in high order of modulation. The suggested algorithm also shows a robustness in reliability and power consumption for both short-packet FD transmissions and high order modulation communications.

Constructions of Nonbinary Codes Meeting the Johnson Bound

Constructions of Nonbinary Codes Meeting the Johnson Bound

Stabilizer quantum codes defined by trace-depending polynomials

Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace [18]. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in [18] and with excellent parameters. Namely, we are able to provide new binary records according to [21] and non-binary codes improving the ones available in the literature.

MeQryEP: A Texture Based Descriptor for Biomedical Image Retrieval.

Image texture analysis is a dynamic area of research in computer vision and image processing, with applications ranging from medical image analysis to image segmentation to content-based image retrieval and beyond. “Quinary encoding on mesh patterns (MeQryEP)” is a new approach to extracting texture features for indexing and retrieval of biomedical images, which is implemented in this work. An extension of the previous study, this research investigates the use of local quinary patterns (LQP) on mesh patterns in three different orientations. To encode the gray scale relationship between the central pixel and its surrounding neighbors in a two-dimensional (2D) local region of an image, binary and nonbinary coding, such as local binary patterns (LBP), local ternary patterns (LTP), and LQP, are used, while the proposed strategy uses three selected directions of mesh patterns to encode the gray scale relationship between the surrounding neighbors for a given center pixel in a 2D image. An innovative aspect of the proposed method is that it makes use of mesh image structure quinary pattern features to encode additional spatial structure information, resulting in better retrieval. On three different kinds of benchmark biomedical data sets, analyses have been completed to assess the viability of MeQryEP. LIDC-IDRI-CT and VIA/I–ELCAP-CT are the lung image databases based on computed tomography (CT), while OASIS-MRI is a brain database based on magnetic resonance imaging (MRI). This method outperforms state-of-the-art texture extraction methods, such as LBP, LQEP, LTP, LMeP, LMeTerP, DLTerQEP, LQEQryP, and so on in terms of average retrieval precision (ARP) and average retrieval rate (ARR).

Weakly Resolvable Block Designs and Nonbinary Codes Meeting the Johnson Bound

Weakly Resolvable Block Designs and Nonbinary Codes Meeting the Johnson Bound

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.

Non-binary QC-LDPC codes for non-Gaussian optical channels

Non-binary QC-LDPC codes for non-Gaussian optical channels

Design of Nonbinary LDPC Cycle Codes with Large Girth from Circulants and Finite Fields

In this paper, we study a class of nonbinary LDPC (NBLDPC) codes whose parity‐check matrices have column weight 2, called NBLDPC cycle codes. We propose a design framework of (2, ρ)‐regular binary quasi‐cyclic (QC) LDPC codes and then construct NBLDPC cycle codes of large girth based on circulants and finite fields by randomly choosing the nonzero field elements in their parity‐check matrices. For enlarging the girth values, our approach is twofold. First, we give an exhaustive search of circulants with column/row weight ρ and design a masking matrix with good cycle distribution based on the edge‐node relation in undirected graphs. Second, according to the designed masking matrix, we construct the exponent matrix based on finite fields. The iterative decoding performances of the constructed codes on the additive white Gaussian noise (AWGN) channel are finally provided.

一种基于有限域的二进制和多进制空间耦合LDPC码构造方法

一种基于有限域的二进制和多进制空间耦合LDPC码构造方法

IMPLEMENTATION OF SOME DECODING ALGORITHMS FOR NB-LDPC CODES ON FPGA

Non-binary low-density parity-check codes (NB-LDPC) provide better error correction performance in comparison with their counterparts. However, the NB-LDPC decoder has a very high complexity, especially the processing of the check node unit. This paper evaluates the error correction performance of some decoding algorithms for NB-LDPC codes in different fields with different codeword lengths. The paper also presents the results of the implementation a decoder structure for the NB-LDPC (35,23) over GF(8) on the Spartan 6 board. Analysis and evaluation results show that decoding quality on hardware is equivalent to simulation results on software, demonstrating high feasibility in implementing decoder on a hardware platform, capable of application in devices of the advanced communication systems or high-speed read-and-write data storages.

Design and Practical Decoding of Full-Diversity Construction A Lattices for Block-Fading Channels

Block-fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. Frequency-hopping schemes and orthogonal frequency division multiplexing (OFDM) can conveniently be modelled as BF channels. Applying lattices in this type of channel entails dividing a lattice point into multiple blocks such that fading is constant within a block but changes, independently, across blocks. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. Recently, the original binary Construction A for lattices, due to Forney, has been generalized to a lattice construction from totally real and complex multiplication (CM) fields. This generalized algebraic Construction A of lattices provides signal space diversity, intrinsically, which is the main requirement for the signal sets designed for fading channels. In this paper, we construct full-diversity algebraic lattices for BF channels using Construction A over totally real number fields. We propose two new decoding methods for these family of lattices which have complexity that grows linearly in the dimension of the lattice. The first decoder is proposed for full-diversity algebraic LDPC lattices which are generalized Construction A lattices with a binary LDPC code as underlying code. This decoding method contains iterative and non-iterative phases. In order to implement the iterative phase of our decoding algorithm, we propose the definition of a parity-check matrix and Tanner graph for full-diversity algebraic Construction A lattices. We also prove that using an underlying LDPC code that achieves the outage probability limit over one-block-fading channel, the constructed algebraic LDPC lattices together with the proposed decoding method admit diversity order $n$ over an $n$ -block-fading channel. Then, we modify the proposed algorithm by removing its iterative phase which enables full-diversity practical decoding of all generalized Construction A lattices without any assumption about their underlying code. In contrast with the known results on AWGN channels in which non-binary Construction A lattices always outperform the binary ones, we provide some instances showing that algebraic Construction A lattices obtained from binary codes outperform the ones based on non-binary codes in block fading channels. Since available lattice construction methods from totally real and complex multiplication (CM) fields do not provide diversity in the binary case, we generalize algebraic Construction A lattices over a wider family of number fields namely monogenic number fields.

BER Staging of Coded-OFDM with Binary and Non-Binary Coding Techniques for LTE Uplink Transmission

BER Staging of Coded-OFDM with Binary and Non-Binary Coding Techniques for LTE Uplink Transmission

Modifications of niederreiter crypto-code construction

Studies of the Niederreiter crypto-code construction at MES have revealed the main reason for the impossibility of practical implementation of decoding algorithms with non-binary codes in the classical scheme. It has been established that fixing a subset of plaintexts is required, for which the error localization procedure cannot be performed with masking selected private matrices X, P, and D. The modified algorithm is developed by shortening the source data and fixing the allowable positional plaintext transformation vectors based on equilibrium coding

ANALYSIS OF THE TRANSMISSION RELIABILITY BETWEEN CONTROL SYSTEMS DEVICES WHEN ERRORS ARE PACKAGED

Context. The reliability indicators of information transmission between control systems elements, taking into account the packaging (grouping) of errors in the communication channel, are analyzed. The research object are the characteristics and parameters of noise stability cyclic binary and non-binary codes for the correction of error packets. The research subject is a theoretical and experimental comparative analysis of the reliability indicators of codes for correcting error packets. Objective. The purpose of the work is to determine the analytical dependencies of the reliability indicators of the selected redundant codes from their parameters and the communication channel properties develop models for their experimental research, formulate recommendations for choosing codes with preset characteristics with certain properties, and an error description model. Methods. The math methods and building theory of noise stability cyclic binary (BCH) and non-binary (Reed-Solomon) codes are used. The analytic formulas for determination of reliability indicators taking into account bits distortions within error packet are received. The dependencies for corrective properties and correct transmission probability form codes and error parameters are researched, the illustrating examples are shown. The simulation circuit design models for control systems with researched methods of noise stability encoding are developed. The experimental research has been done, based on the results, conclusions and recommendations for choice of code parameters for preset reliability indicators for providing of maximum efficiency (information rate) are made. Results. The dependencies of the reliability indicators (correct transmission probability) and the corrective properties (the length of corrected errors packet) from its parameters (the number of redundancy symbols, the interleaving degree – for BCH-codes; the Galua field module, number of corrected errors – for Reed-Solomon codes) are received. The recommendations of using the received results for code parameters choice are given. Conclusions. The performed researches allows calculate and reasonably choose of redundancy codes parameters for preset reliability indicators и the behavior model of errors within the communication channel. It makes possible to design and implement reliable control systems with preset reliability indicators and maximum information rate.

Pergroup and Joint Optimization of Max-Dmin Precoder for MIMO with LDPC Coding Using QAM Modulation

MIMO technology not only offers diversity and capacity gains, but also provides higher spectral efficiency and significant link reliability over SISO systems. Many methods are developed to exploit the diversity offered by multi-antenna systems such as Alamouti code and spatial multiplexing that do not require transmitter-side channel status information (Tx-CSI). Other power allocation optimization techniques, also known as precoding, require a full or partial Tx-CSI. These techniques aim to transform the signal before transmission according to different criteria, among which the minimal Euclidean distance seems to be very effective and continues to interest the researchers. Given perfect channel state information at both sides of the communication, we propose in this paper a novel design of wireless transmission schemes that joint the minimal Euclidean distance precoder and error correction coding based on the non-binary low-density parity-check code (NB-LDPC), to finally determine a power allocation optimization solution that adapts a linear precoding block to an NB-LDPC encoded MIMO transmission. In this paper we use a quadrature amplitude modulation (QAM), over a Rayleigh fading channel with a maximum likelihood detection. Simulations results in term of bit error rate confirmed that NB-LDPC codes are particularly well suited to be jointly used with precoding schemes based on the maximization of the minimum Euclidean distance criterion.

Evaluation of ccomputational complexity and shortcomings of non-binary low density decoding algorithms

Operational requirements for processing, receiving and transmitting digital information devices constantly increase, due to the need to attract significant resources, especially intellectual. This is caused by the necessity to perform the tasks that are aimed at improving the efficiency of tools, methods and specific algorithms that implement processing, receiving and transmitting information technologies. One of the most dynamically developing approaches for solving the above problem is the coding theory, in particular noise-resistant coding. As far as the energy costs are concerned, the most effective noise-resistant codes are turbo codes and the codes with a low density of parity checks. The paper reviews the algorithms that allow the decoding of non-binary codes with a low density of parity, as well as the distribution algorithm and its modifications. Their significant drawbacks and major modifications applied to eliminate them are briefly overviewed in the paper; estimates of the computational complexity of the algorithms are given.

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.

Performances of the Decoding Algorithms near Shannon Limit

Forward error correction technique based on the optimization procedures for multithreshold decoding algorithms is considered. Recent advances as well as new opportunities of the multithreshold decoders for use in the systems working at the Shannon’s limit are analysed. Performances of the multithreshold decoders for self-orthogonal binary and non-binary codes over Gaussian channels are presented. Simulation results of BER performance of proposed decoder is shown to be close to the results provided by optimum total search methods.

Local equivalence of qudit color codes and toric codes

In this paper, we study nonbinary toric codes and color codes. While the relationships between the qubit color codes and toric codes have been studied extensively from various perspectives, similar studies for qudit toric codes and color codes are missing. We show that just as in the binary case, qudit color codes can be mapped to qudit toric codes. Qudit color codes are known to have the capability for implementing the generalized Clifford group transversally. The proposed mapping of qudit color codes to toric codes would allow us to port the computational capabilities of the qudit color codes to qudit toric codes. Another application would be to decode qudit color codes via qudit toric codes. Along the way, we show an important structural result for higher-dimensional qudit toric codes. Specifically, we show that the toric codes over prime power dimension can be viewed as a direct sum of copies of codes over the prime dimension. This result simplifies the study of codes over prime power dimension by reducing them to the study of codes over the prime dimension. Furthermore, we provide explicit circuits for the transformation between qudit color codes and toric codes.

Generalization of IPP Codes and IPP Set Systems

A quarter century ago Chor, Fiat, and Naor proposed mathematical models for revealing a source of illegal redistribution of digital content (tracing traitors) in the broadcast encryption framework, including the following two combinatorial models: nonbinary IPP codes, based on an (n, n)-threshold secret sharing scheme, and IPP set systems, based on the general (w, n)-threshold secret sharing scheme. We propose a new scheme combining the main ideas of nonbinary IPP codes and IPP set systems, which can also be considered as a generalization of nonbinary IPP codes to the case of constant-weight codes. In the simplest case of a coalition of size two, we compare the new scheme with previously known ones.