Linear Codes Research Articles

On Linear Codes over Finite Singleton Local Rings

The study of linear codes over local rings, particularly non-chain rings, imposes difficulties that differ from those encountered in codes over chain rings, and this stems from the fact that local non-chain rings are not principal ideal rings. In this paper, we present and successfully establish a new approach for linear codes of any finite length over local rings that are not necessarily chains. The main focus of this study is to produce generating characters, MacWilliams identities and generator matrices for codes over singleton local Frobenius rings of order 32. To do so, we first start by characterizing all singleton local rings of order 32 up to isomorphism. These rings happen to have strong connections to linear binary codes and Z4 codes, which play a significant role in coding theory.

Obtaining new classes of optimal linear codes by puncturing and shortening optimal cyclic codes

AbstractIn this paper we use the puncturing and shortening techniques on two already-known classes of optimal cyclic codes in order to obtain three new classes of optimal linear codes achieving the Griesmer bound. The weight distributions for these codes are settled. We also investigate their dual codes and show that they are either optimal or almost optimal with respect to the sphere-packing bound. Moreover, these duals contain classes of almost maximum distance separable codes which are shown to be proper for error detection. Further, some of the obtained optimal linear codes are suitable for constructing secret sharing schemes with nice access structures.

Emotion Modeling in Speech Signals: Discrete Wavelet Transform and Machine Learning Tools for Emotion Recognition System

Speech emotion recognition (SER) is a challenging task due to the complex and subtle nature of emotions. This study proposes a novel approach for emotion modeling using speech signals by combining discrete wavelet transform (DWT) with linear prediction coding (LPC). The performance of various classifiers, including support vector machine (SVM), K-Nearest Neighbors (KNN), Efficient Logistic Regression, Naive Bayes, Ensemble, and Neural Network, was evaluated for emotion classification using the EMO-DB dataset. Evaluation metrics such as area under the curve (AUC), average prediction accuracy, and cross-validation techniques were employed. The results indicate that KNN and SVM classifiers exhibited high accuracy in distinguishing sadness from other emotions. Ensemble methods and Neural Networks also demonstrated strong performance in sadness classification. While Efficient Logistic Regression and Naive Bayes classifiers showed competitive performance, they were slightly less accurate compared to other classifiers. Furthermore, the proposed feature extraction method yielded the highest average accuracy, and its combination with formants or wavelet entropy further improved classification accuracy. On the other hand, Efficient Logistic Regression exhibited the lowest accuracies among the classifiers. The uniqueness of this study was that it investigated a combined feature extraction method and integrated them to compare with various forms of combinations. However, the purposes of the investigation include improved performance of the classifiers, high effectiveness of the system, and the potential for emotion classification tasks. These findings can guide the selection of appropriate classifiers and feature extraction methods in future research and real-world applications. Further investigations can focus on refining classifiers and exploring additional feature extraction techniques to enhance emotion classification accuracy.

Large (k,3)-arcs in PG(2,19) and the related linear codes

A (n,r)-arc in a projective plane PG(2,q) is a set of n points such that some r, but no r+1 of them, are collinear. A (n,r)-arc is called complete if it is not contain in a (n+1,r)-arc. A linear -code over a finite field is a k-dimensional subspace of with minimum hamming distance d and length n. A code with parameters with Griesmer bound , is called Griesmer code. The major aim of this research is to find large size for the complete (k,3)-arcs in the projective plane of order nineteen PG(2,19) using the method of secants distributions, and the disjoint union of arcs, as well as, adding and removing points to (from) particular conic respectively. Also, we find the Griesmer codes that correspond to each large complete (k,3)-arcs, k=29,30,31. We introduced 20 inequivalent (29,3)-arcs up to secant distribution, 10 of them are complete. Also, we introduced 8 inequivalent (30,3)-arcs up to secant distribution, 2 of them are complete. Moreover, we construct 3 inequivalent complete (31,3)-arcs up to secant distribution, and then find the corresponding linear codes to some of the (29,3)-arcs, (30,3)-arcs and (31,3)-arcs. In particular, we established 3 types of Griesmer codes, and we find the weight enumerator that correspond to each one of them.

Private information retrieval from locally repairable databases with colluding servers

We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups g, locality r, local distance δ and dimension k. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed–Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only r nodes (out of r+δ−1) are downloaded per server. The PIR scheme achieves the (download) rate R=(N−k−rt+1)/N, where N=gr is the length of the MDS code obtained after removing the local parities, and for any t colluding servers such that k+rt≤N. For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed–Solomon codes, but Reed–Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications.

Design of Universal Code Generator for Multi-Constellation Multi-Frequency GNSS Receiver

A multi-constellation, multi-frequency Global Navigation Satellite System (GNSS) receiver is capable of simultaneously receiving signals from multiple satellite constellations across various frequency bands. This allows for increased observations, thereby enhancing navigation accuracy, continuity, effectiveness, and reliability. The spread spectrum code structures used in satellite navigation signals differ among systems. Compatible code generators are employed in multi-constellation, multi-frequency GNSS receivers to support tasks such as signal acquisition and tracking. There are three main types of spread spectrum code structures: Linear Feedback Shift Register (LFSR), Legendre sequences, and Memory codes. The Indian Regional Navigation Satellite System (IRNSS) released the L1-SPS (Standard Positioning Service) signal format in August 2023, which utilizes the Interleaved Z4-linear ranging code (IZ4 code) as its spread spectrum code. Currently, there is no universal code generator design compatible with the IZ4 code. In this paper, a proposed universal code generator is based on the hardware structure of the IRNSS IZ4 code generator. It achieves compatibility with all LFSR-based spread spectrum codes and enables parallel generation of multiple sets of GNSS signal spread spectrum codes, thereby improving hardware utilization efficiency. The proposed structure is implemented and validated using FPGA design, and resource consumption is provided as part of the validation results.

The investigation and estimation of flexural strength in shear-critically designed RC beams considering the steel fiber content

Despite numerous studies investigating the effectiveness of steel fibers in RC members subjected to various loading conditions, the majority ignored the impact of the interaction between the tensile reinforcement ratio, compressive strength of fibrous concrete, and steel fiber dosage. This paper aimed to inspect and quantify the effects of hooked-end steel fiber dosage on the bending behavior of RC beams considering the interaction arising from various longitudinal reinforcement ratios. For this purpose, an extensive and hybrid investigation involving both experimental and numerical techniques was conducted. In the experimental part, an investigation was carried out to evaluate the contribution of a specific amount of fiber ratio and various aspect ratios to the behavior of large-scale RC beams having a shear-deficient design. Three beams with a 0.5% hooked-end steel fiber ratio and one without fibers but with a minimum amount of stirrups were produced, and three-point bending tests were conducted. All beams having fibers utilized full flexural strength as in the beam with stirrups. Once the experimental studies were completed, the numerical models were then developed and validated with those of experimental three-point bending tests using the non-linear finite element code VecTor2 to establish a basis for the comprehensive parametric study. The parametric study consisted of 33 analyses. Rebar and volumetric fiber ratios were selected to be variables, and the ratios adopted were within the typical range used in structural engineering applications. Based on the regression analysis a linear and a quadratic model were proposed for estimating the yield and peak strength depending on the steel fiber content. Finally, the obtained models were verified using work from the literature and found to be effective in forecasting the strength of shear-deficient RC beams with 0.8%, 1.0%, and 1.2% tensile reinforcement ratios and hooked end steel fibers (0.5 to 1.5%).

Imperceptible and Reversible Acoustic Watermarking Based on Modified Integer Discrete Cosine Transform Coefficient Expansion

This paper aims to explore an alternative reversible digital watermarking solution to guarantee the integrity of and detect tampering with data of probative importance. Since the payload for verification is embedded in the contents, algorithms for reversible embedding and extraction, imperceptibility, payload capacity, and computational time are issues to evaluate. Thus, we propose a reversible and imperceptible audio information-hiding algorithm based on modified integer discrete cosine transform (intDCT) coefficient expansion. In this work, the original signal is segmented into fixed-length frames, and then intDCT is applied to each frame to transform signals from the time domain into integer DCT coefficients. Expansion is applied to DCT coefficients at a higher frequency to reserve hiding capacity. Objective evaluation of speech quality is conducted using listening quality objective mean opinion (MOS-LQO) and the segmental signal-to-noise ratio (segSNR). The audio quality of different frame lengths and capacities is evaluated. Averages of 4.41 for MOS-LQO and 23.314 [dB] for segSNR for 112 ITU-T test signals were obtained with a capacity of 8000 bps, which assured imperceptibility with the sufficient capacity of the proposed method. This shows comparable audio quality to conventional work based on Linear Predictive Coding (LPC) regarding MOS-LQO. However, all segSNR scores of the proposed method have comparable or better performance in the time domain. Additionally, comparing histograms of the normalized maximum absolute value of stego data shows a lower possibility of overflow than the LPC method. A computational cost, including hiding and transforming, is an average of 4.884 s to process a 10 s audio clip. Blind tampering detection without the original data is achieved by the proposed embedding and extraction method.

Modified Joint Source–Channel Trellises for Transmitting JPEG Images over Wireless Networks

This paper presents a joint source–channel image transmission model based on a modified trellis construction using variable-length and fixed-length codes. The model employs linear block trellis codes and a modified Bahl–Cocke–Jelinek–Raviv algorithm for decoding in the source channel. Additionally, this model utilizes a hierarchical tree and a parity-check matrix to combine the source and channel codes, reducing errors in the decoding process and enabling the decoding of longer sequences without substantial loss of quality.

Cyclic Codes over a Non-Local Non-Unital Ring

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H=⟨a,b∣2a=2b=0,a2=0,b2=b,ab=ba=0⟩. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2n over F2 is studied.

Weight distribution of random linear codes and Krawtchouk polynomials

AbstractFor and pick uniformly at random vectors in and let be the orthogonal complement of their span. Given with , let be the random variable that counts the number of words in of Hamming weight (where is assumed to be an even integer). Linial and Mosheiff [Random Struct. Algorithms. 62 (2023), 68Ű‐130] determined the asymptotics of the moments of of all orders . In this paper we extend their estimates up to moments of linear order. Our key observation is that the behavior of the suitably normalized moment of is essentially determined by the norm of the Krawtchouk polynomial .

Linear Codes Correcting Repeated Bursts Equipped with Homogeneous Distance

Linear Codes Correcting Repeated Bursts Equipped with Homogeneous Distance

Linear codes associated to determinantal varieties in the space of hermitian matrices

Linear codes associated to determinantal varieties in the space of hermitian matrices

Novel compressed linear network coding vectors for multihop communication networks

Novel compressed linear network coding vectors for multihop communication networks

New Constructions of Optimal Linear Codes From Simplicial Complexes

New Constructions of Optimal Linear Codes From Simplicial Complexes

Can using a pre-trained deep learning model as the feature extractor in the bag-of-deep-visual-words model always improve image classification accuracy?

This article investigates whether higher classification accuracy can always be achieved by utilizing a pre-trained deep learning model as the feature extractor in the Bag-of-Deep-Visual-Words (BoDVW) classification model, as opposed to directly using the new classification layer of the pre-trained model for classification. Considering the multiple factors related to the feature extractor -such as model architecture, fine-tuning strategy, number of training samples, feature extraction method, and feature encoding method-we investigate these factors through experiments and then provide detailed answers to the question. In our experiments, we use five feature encoding methods: hard-voting, soft-voting, locally constrained linear coding, super vector coding, and fisher vector (FV). We also employ two popular feature extraction methods: one (denoted as Ext-DFs(CP)) uses a convolutional or non-global pooling layer, and another (denoted as Ext-DFs(FC)) uses a fully-connected or global pooling layer. Three pre-trained models-VGGNet-16, ResNext-50(32×4d), and Swin-B-are utilized as feature extractors. Experimental results on six datasets (15-Scenes, TF-Flowers, MIT Indoor-67, COVID-19 CXR, NWPU-RESISC45, and Caltech-101) reveal that compared to using the pre-trained model with only the new classification layer re-trained for classification, employing it as the feature extractor in the BoDVW model improves the accuracy in 35 out of 36 experiments when using FV. With Ext-DFs(CP), the accuracy increases by 0.13% to 8.43% (averaged at 3.11%), and with Ext-DFs(FC), it increases by 1.06% to 14.63% (averaged at 5.66%). Furthermore, when all layers of the pre-trained model are fine-tuned and used as the feature extractor, the results vary depending on the methods used. If FV and Ext-DFs(FC) are used, the accuracy increases by 0.21% to 5.65% (averaged at 1.58%) in 14 out of 18 experiments. Our results suggest that while using a pre-trained deep learning model as the feature extractor does not always improve classification accuracy, it holds great potential as an accuracy improvement technique.

The c-Differential-Linear Connectivity Table of Vectorial Boolean Functions.

Vectorial Boolean functions and codes are closely related and interconnected. On the one hand, various requirements of binary linear codes are needed for their theoretical interests but, more importantly, for their practical applications (such as few-weight codes or minimal codes for secret sharing, locally recoverable codes for storage, etc.). On the other hand, various criteria and tables have been introduced to analyse the security of S-boxes that are related to vectorial Boolean functions, such as the Differential Distribution Table (DDT), the Boomerang Connectivity Table (BCT), and the Differential-Linear Connectivity Table (DLCT). In previous years, two new tables have been proposed for which the literature was pretty abundant: the c-DDT to extend the DDT and the c-BCT to extend the BCT. In the same vein, we propose extended concepts to study further the security of vectorial Boolean functions, especially the c-Walsh transform, the c-autocorrelation, and the c-differential-linear uniformity and its accompanying table, the c-Differential-Linear Connectivity Table (c-DLCT). We study the properties of these novel functions at their optimal level concerning these concepts and describe the c-DLCT of the crucial inverse vectorial (Boolean) function case. Finally, we draw new ideas for future research toward linear code designs.

Matrix product and quasi-twisted codes in one class

Many classical constructions, such as Plotkin’s and Turyn’s, were generalized by matrix product (MP) codes. Quasi-twisted (QT) codes, on the other hand, form an algebraically rich structure class that contains many codes with best-known parameters. We significantly extend the definition of MP codes to establish a broader class of generalized matrix product (GMP) codes that contains QT codes as well. We propose a generator matrix formula for any linear GMP code and provide a condition for determining the code size. We prove that any QT code has a GMP structure. Then we show how to build a generator polynomial matrix for a QT code from its GMP structure, and vice versa. Even though the class of QT codes contains many codes with best-known parameters, we present different examples of GMP codes with best-known parameters that are neither MP nor QT. Two different lower bounds on the minimum distance of GMP codes are presented; they generalize their counterparts in the MP codes literature. The second proposed lower bound replaces the non-singular by columns matrix with a less restrictive condition. Some examples are provided for comparing the two proposed bounds, as well as showing that these bounds are tight.

New Golay decoding method using auto-encoder and OSD

In this study, we explore the potential of leveraging machine learning techniques, specifically auto-encoders (AE), for the decoding of linear block codes. Our findings suggest that this approach can outperform the conventional ordered statistics decoding (OSD) method, especially in a Rayleigh fading channel environment. We have rigorously trained the AE under both additive white Gaussian noise and Rayleigh fading channel conditions to ensure robustness in its performance. The output of the AE is combined with the received vector in a suitable manner to perform OSD. Through our experiments, we demonstrate that this proposed decoding approach yields better results than the conventional OSD method in Rayleigh fading channel when we used (23,12) Golay code.

On linear codes with random multiplier vectors and the maximum trace dimension property

Abstract Let C C be a linear code of length n n and dimension k k over the finite field F q m {{\mathbb{F}}}_{{q}^{m}} . The trace code Tr ( C ) {\rm{Tr}}\left(C) is a linear code of the same length n n over the subfield F q {{\mathbb{F}}}_{q} . The obvious upper bound for the dimension of the trace code over F q {{\mathbb{F}}}_{q} is m k mk . If equality holds, then we say that C C has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let C a {C}_{{\boldsymbol{a}}} denote the code obtained from C C and a multiplier vector a ∈ ( F q m ) n {\boldsymbol{a}}\in {\left({{\mathbb{F}}}_{{q}^{m}})}^{n} . In this study, we give a lower bound for the probability that a random multiplier vector produces a code C a {C}_{{\boldsymbol{a}}} of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever n ≥ m ( k + h ) n\ge m\left(k+h) , where h ≥ 0 h\ge 0 is the Singleton defect of C C . For the extremal case n = m ( h + k ) n=m\left(h+k) , numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank.