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The paper lists the main open questions left in the area both theoretically and practically, mainly related with the inherent tension between communication efficiency and quality of assumptions in proof systems. This special issue of Designs, Codes and Cryptography on the topic Mathematics of Zero-Knowledge aims at encouraging mathematicians and computer scientists with a strong interest in Mathematics to contribute to the burgeoning topic of zero-knowledge proofs. [Extracted from the article]

Purpose: There is a growing need for methodological plurality in advancing operations management (OM), especially with the emergence of machine learning (ML) techniques for analysing extensive textual data. To bridge this knowledge gap, this paper introduces a new methodology that combines ML techniques with traditional qualitative approaches, aiming to reconstruct knowledge from existing publications. Design/methodology/approach: In this pragmatist-rooted abductive method where human-machine interactions analyse big data, the authors employ topic modelling (TM), an ML technique, to enable constructivist grounded theory (CGT). A four-step coding process (Raw coding, expert coding, focused coding and theory building) is deployed to strive for procedural and interpretive rigour. To demonstrate the approach, the authors collected data from an open-source professional project management (PM) website and illustrated their research design and data analysis leading to theory development. Findings: The results show that TM significantly improves the ability of researchers to systematically investigate and interpret codes generated from large textual data, thus contributing to theory building. Originality/value: This paper presents a novel approach that integrates an ML-based technique with human hermeneutic methods for empirical studies in OM. Using grounded theory, this method reconstructs latent knowledge from massive textual data and uncovers management phenomena hidden from published data, offering a new way for academics to develop potential theories for business and management studies. [ABSTRACT FROM AUTHOR]

Many papers have been devoted to applying fuzzy sets to algebraic structures. In this paper, based on ideals, we investigate residuated lattices from fuzzy set theory, lattice theory, and coding theory points of view, and some applications of fuzzy sets in residuated lattices are presented. Since ideals are important concepts in the theory of algebraic structures used for formal fuzzy logic, first, we investigate the lattice of fuzzy ideals in residuated lattices and study some connections between fuzzy sets associated to ideals and Hadamard codes. Finally, we present applications of fuzzy sets in coding theory. [ABSTRACT FROM AUTHOR]

We investigate CSS and CSS-T quantum error-correcting codes from the point of view of their existence, rarity, and performance. We give a lower bound on the number of pairs of linear codes that give rise to a CSS code with good correction capability, showing that such pairs are easy to produce with a randomized construction. We then prove that CSS-T codes exhibit the opposite behaviour, showing also that, under very natural assumptions, their rate and relative distance cannot be simultaneously large. This partially answers an open question on the feasible parameters of CSS-T codes. We conclude with a simple construction of CSS-T codes from Hermitian curves. The paper also offers a concise introduction to CSS and CSS-T codes from the point of view of classical coding theory. [ABSTRACT FROM AUTHOR]

In this paper, three discrete transforms related to vector spaces over finite fields are studied. For our purposes, and according to the properties of the finite fields, the most suitable transforms are as follows: for binary fields, this is the Walsh–Hadamard transform; for odd prime fields, the Vilenkin–Chrestenson transform; and for composite fields, the trace transform. A factorization of the transform matrices using Kronecker power is given so that the considered discrete transforms are reduced to the fast discrete transforms. Examples and applications are also presented of the considered transforms in coding theory for calculating the weight distribution of a linear code. [ABSTRACT FROM AUTHOR]

Cloud storage is an indispensable part of cloud computing solutions and the security of its stored data has become a key issue in the research and application of cloud storage systems. To solve this problem, this paper studies the anti-eavesdropping regenerating code technology for cloud storage systems, from the perspective of information theory. As opposed to the existing research ideas on regenerating code theory, that enable the system to obtain strong/weak security, this paper focuses on quantifying the relationship between security and system performance parameters, evaluating the system performance gains that can be obtained by appropriately reducing security, and designing regenerating code schemes with different information security levels to meet the personalized requirements of cloud storage customers. This paper puts forward a generalized matrix transposing method and applies it to the coding construction of fractional repetition codes. The scheme proposed in this paper will provide new ideas and methods for research on secure regenerating code technology in cloud storage systems. [ABSTRACT FROM AUTHOR]

The predictive coding theory is currently widely accepted as the theoretical basis of perception and chronic perceptual disorders are explained as the maladaptive compensation of the brain to a prediction error. Although this gives us a general framework to work with, it is still not clear who may be more susceptible and/or vulnerable to aberrations in this system. In this paper, we study changes in predictive coding through the lens of tinnitus and pain. We take a step back to understand how the predictive coding system develops from infancy, what are the different neural and bio markers that characterise this system in the acute, transition and chronic phases and what may be the factors that pose a risk to the aberration of this system. Through this paper, we aim to identify people who may be at a higher risk of developing chronic perceptual disorders as a reflection of aberrant predictive coding, thereby giving future studies more facets to incorporate in their investigation of early markers of tinnitus, pain and other disorders of predictive coding. We therefore view this paper to encourage the thinking behind the development of preclinical biomarkers to maladaptive predictive coding. [ABSTRACT FROM AUTHOR]

In this paper, we firstly introduce the Gaussian Leonardo polynomial sequences {GLen(X)}⃵n=0 and we obtain Binet's formula, generating function of this sequence. Moreover, we define the matrix GL(X) in the form of 3 x 3. Finally, we study on the coding and decoding applications of the Leonardo number by using the Leonardo matrix. [ABSTRACT FROM AUTHOR]

In this paper, we first consider the existence conditions, the construction and the enumeration of Hermitian self-dual θ -cyclic and θ -negacyclic codes over I F p 2 , where p is a prime number and θ is the Frobenius automorphism over I F p 2 . We then give necessary and sufficient conditions for the existence of Hermitian self-dual θ -cyclic and θ -negacyclic codes over I F p e where e is an even integer greater than 2. [ABSTRACT FROM AUTHOR]

In the paper, we explore the simplex and the MacDonald codes over the finite ring ZpR1R2. Our investigation focuses on the unique properties of these codes, with the particular attention to their weight distributions and Gray images. The weight distribution is a crucial aspect as it provides insights into the error-detection and error-correction capabilities of the codes. Gray images play a significant role in understanding the structure and behavior of these codes. By examining the dual Gray images of simplex and MacDonald codes over ZpR1R2, we aim to develop efficient secret sharing schemes. These schemes benefit from the inherent properties of the codes, such as minimal weight and redundancy, which are essential for secure and reliable information sharing. Understanding the access structure of these schemes is vital, as it determines which subsets of participants can reconstruct the secret. Our study draws on various properties to elucidate this access structure, ensuring that the schemes are secure and efficient. Through this comprehensive analysis, we contribute to the field of coding theory by demonstrating how simplex and MacDonald codes over ZpR1R2 can be effectively utilized in cryptographic applications, particularly in designing robust and reliable secret sharing mechanisms. [ABSTRACT FROM AUTHOR]

In general coding theory, we often assume that error is observed in transferring or storing encoded symbols, while the process of encoding itself is error-free. Motivated by recent applications of coding theory, in this paper, we consider the case where the process of encoding is distributed and prone to error. We introduce the problem of distributed encoding, comprised of a set of $K \in \mathbb {N}$ isolated source nodes and $N \in \mathbb {N}$ encoding nodes. Each source node has one symbol from a finite field, which is sent to each of the encoding nodes. Each encoding node stores an encoded symbol from the same field, as a function of the received symbols. However, some of the source nodes are controlled by the adversary and may send different symbols to different encoding nodes. Depending on the number of the adversarial nodes, denoted by $\beta \in \mathbb {N}$ , and the cardinality of the set of symbols that each one generates, denoted by $v \in \mathbb {N}$ , the process of decoding from the encoded symbols could be impossible. Assume that a decoder connects to an arbitrary subset of $t \in \mathbb {N}$ encoding nodes and wants to decode the symbols of the honest nodes correctly, without necessarily identifying the sets of honest and adversarial nodes. An important characteristic of a distributed encoding system is $t^{*} \in \mathbb {N}$ , the minimum of such $t$ , which is a function of $K$ , $N$ , $\beta $ , and $v$. In this paper, we study the distributed linear encoding system, i.e. one in which the encoding nodes use linear coding. We show that $t^{*}_{\textrm {Linear}}=K+2\beta (v-1)$ , if $N\ge K+2\beta (v-1)$ , and $t^{*}_{\textrm {Linear}}=N$ , if $N\le K+2\beta (v-1)$. In order to achieve $t^{*}_{\textrm {Linear}}$ , we use random linear coding and show that in any feasible solution that the decoder finds, the messages of the honest nodes are decoded correctly. In order to prove the converse of the fundamental limit, we show that when the adversary behaves in a particular way, it can always confuse the decoder between two feasible solutions that differ in the message of at least one honest node. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show the potential value of using the constructivist grounded theory (CGT) methodology in the Urban and Regional Planning discipline. This is done by reflecting on the application of the CGT fundamentals defined by Kathy Charmaz, applied in a doctoral thesis in Urban and Regional Planning. The substantive area of research in which the thesis took place was the practical training experience of planners in becoming professionally registered. The thesis collected primary data through interviews with 14 planners who have undergone the process of practical training to record, document, and analyse their experiences as a prerequisite and requirement of professional registration. The use of the CGT methodology provides valuable insight into the registration process of planners in South Africa and encourages doctoral candidates to use CGT methodology with the aim of generating new knowledge and theory in the built environment. This paper recommends that CGT methodology should be encouraged, particularly when contributing to fields that are under-researched, limited, or non-existent. [ABSTRACT FROM AUTHOR]

Resource misallocation (RM) refers to the existence of marginal output inequalities between different industries or companies in an economy. Prior studies of RM have mostly focused on effect analysis, construction industry structure upgrades, and organization management. However, these studies have been fragmented and unrelated. This paper analyzes the status quo, consequences, and emerging trends of RM research at the macroscopic level based on current problems and with the aim of exploring potential solutions. Drawing on grounded theory, a qualitative analysis using text-mining is used to analyze the characteristics of 124 RM-related papers. The results more comprehensively and systematically reveal that current RM research encompasses four major dimensions of sources and concepts, misallocation degree measurement and characterization, focused issues (field), and RM research deficiencies. Methods for measuring RM have also been developed from the simple proportional method to current mainstream methods (e.g., growth rate decomposition and variant substitution). We conclude that, in order for this discipline to thrive and effectively reduce RM, future research into RM should focus on core categories, especially the reform of market-oriented factors, transformation of government functions, construction industrial structure adjustment, and methods of income distribution. This systematic review provides a discipline oversight and uncovers necessary and potential research directions. [ABSTRACT FROM AUTHOR]

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 Z 4 codes, which play a significant role in coding theory. [ABSTRACT FROM AUTHOR]

As an algorithmic framework, message passing is extremely powerful and has wide applications in the context of different disciplines including communications, coding theory, statistics, signal processing, artificial intelligence and combinatorial optimization. In this paper, we investigate the performance of a message-passing algorithm called min-sum belief propagation (BP) for the vertex-disjoint shortest $k$ -path problem ($k$ -VDSP) on weighted directed graphs, and derive the iterative message-passing update rules. As the main result of this paper, we prove that for a weighted directed graph $G$ of order $n$ , BP algorithm converges to the unique optimal solution of $k$ -VDSP on $G$ within $O(n^{2}w_{max})$ iterations, provided that the weight $w_{e}$ is nonnegative integral for each arc $e\in E(G)$ , where $w_{max}=\max \{w_{e}: e\in E(G)\}$. To the best of our knowledge, this is the first instance where BP algorithm is proved correct for NP-hard problems. Additionally, we establish the extensions of $k$ -VDSP to the case of multiple sources or sinks. [ABSTRACT FROM AUTHOR]

In this paper we settle the question of whether a finite-dimensional vector space V over F p , with p an odd prime, and the family of all the k-sets of elements of V summing up to a given element x, form a 1- (v , k , λ 1) or a 2- (v , k , λ 2) block design, and, in either case, we find a closed form for λ i and characterize the automorphism group. The question is discussed also in the case where the elements of the k-sets are required to be all nonzero, as the two cases happen to be intrinsically inseparable. The "twin case" p = 2 , which has strict connections with coding theory, was completely discussed in a recent paper by G. Falcone and the present author. [ABSTRACT FROM AUTHOR]

Energy efficiency (EE) is an important indicator in ultra‐reliable and low‐latency communication (URLLC). Power allocation is considered as an effective method to achieve high EE in URLLC. However, since the EE optimization problem is non‐convex, it is difficult to obtain the analytical solution efficiently. Moreover, to ensure reliable and low‐latency communication within a finite blocklength, the Shannon formula becomes impractical for URLLC. Therefore, finite blocklength coding theory is used to meet the requirements of URLLC. In this paper, the EE problem of URLLC is formulated and the power allocation function is parameterized to be optimized through a deep neural network (DNN). The DNN is trained through the primal‐dual iterative algorithm offline in the unsupervised manner, and can be deployed online to achieve real time power allocation results. The numerical results show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Fog computing and the Internet of Things (IoT) played a crucial role in storing data in the third-party server. Fog computing provides various resources to collect data by managing data security. However, intermediate attacks and data sharing create enormous security challenges like data privacy, confidentiality, authentication, and integrity issues. Various researchers introduce several cryptographic techniques; security is still significant while sharing data in the distributed environment. Therefore, in this paper, Code-Based Encryption with the Energy Consumption Routing Protocol (CBE-ECR) has been proposed for managing data security and data transmission protocols using keyed-hash message authentication. Initially, the data have been analyzed, and the distributed cluster head is selected, and the stochastically distributed energy clustering protocol is utilized for making the data transmission. Code-driven cryptography relies on the severity of code theory issues such as disorder demodulation and vibration required to learn equivalence. These crypto-systems are based on error codes to build a single-way function. The encryption technique minimizes intermediate attacks, and the data have protected all means of transmission. In addition to data security management, the introduced CBE-ECR reduces unauthorized access and manages the network lifetime successfully, leading to the effective data management of 96.17% and less energy consumption of 21.11% than other popular methods.The effectiveness of the system is compared to the traditional clustering techniques. [ABSTRACT FROM AUTHOR]

Binary linear codes with few weights have wide applications in communication, secret sharing schemes, authentication codes, association schemes, strongly regular graphs, etc. Projective binary linear codes are among the most important subclasses of binary linear codes for practical applications. In this paper, motivated by the two excellent recent papers (Li et al. in IEEE Trans Inf Theory 67(7):4263–4275, 2021) and (Wang et al. in IEEE Trans Inf Theory 67(8):5133–5148, 2021), several new families of few-weight projective binary linear codes are constructed from the defining sets, and then their Hamming weight distributions are determined by employing the Walsh transform of the corresponding two-to-one functions over finite fields with the even characteristic. Our constructions can produce binary linear codes with new parameters. Some of the constructed binary linear codes are optimal or almost optimal according to the online Database of Grassl, and the duals of some of them are distance-optimal with respect to the sphere packing bound. This paper also shows once again that the two-to-one functions initially studied in (Mesnager and Qu in IEEE Trans Inf Theory 65(12):7884–7895, 2019) are also promising objects in coding theory. Although our derived codes use objects considered in the very recent literature, the analysis of our designed codes involves functions having different algebraic structures (and, therefore, other Walsh transform distribution) and requires solving new systems of equations over finite fields, which is an essential step in determining the weight distribution of our constructed codes. As applications, some of the codes presented in this paper can be used to construct association schemes and secret sharing schemes with interesting access structures. [ABSTRACT FROM AUTHOR]

In recent years, soft sets have been widely used in many important decisionmaking real-life problems. In this paper, observing the usage of soft sets in such kind of vital problems, we have introduced the bijective-unitary bijective soft rings. Firstly, we have defined and exemplified a bijective soft ring and a unitary bijective soft ring. Moreover, we have presented some applications of bijective soft rings. We have shown the usage of bijective soft rings in coding theory. In this context, we have observed that by obtaining a bijective soft ring over a finite ring, we have a coding matrix to encode a given set of messages. Besides these applicable results, we have also obtained some relations between bijective soft and classical rings. [ABSTRACT FROM AUTHOR]

Binary codes have a special place in coding theory since they are one of the most commonly used in practice. There are classes of codes specific only to the binary case. One such class is self-complementary codes. Self-complementary linear codes are binary codes that, together with any vector, contain its complement as well. This paper is about binary linear self-complementary codes. A natural goal in coding theory is to find a linear code with a given length n and dimension k such that the minimum distance d is maximal. Codes with these properties are called optimal. Another important issue is classifying the optimal codes, i.e., finding exactly one representative of each equivalence class. In some sense, the classification problem is more general than the minimum distance bounds problem. In this work, we summarize the classification results for self-complementary codes with the maximum possible minimum distance and a length of up to 20. For the classification, we developed a new algorithm that is much more efficient compared to existing ones in some cases. [ABSTRACT FROM AUTHOR]

In the fields of combinatorics on words and theory of codes, a two-word language { x , y } is a code if and only if x y ≠ y x . But up to now, corresponding characterizations for a three-word language, which forms a code, have not been completely found. Let X = { x , y , z } be a three-word language and | x | , | y | , | z | be their lengths. When | x | = | y | < | z | , a necessary and sufficient condition for X to be a code was obtained in 2018. If | x | < | y | = | z | ≤ 2 | x | , a necessary and sufficient condition for X to be a code is proposed in this paper. [ABSTRACT FROM AUTHOR]

Sunflower in coding theory is a class of important subspace codes and can be used to construct linear codes. In this paper, we study the minimality of linear codes over F q constructed from sunflowers of size s in all cases. For any sunflower, the corresponding linear code is minimal if s ≥ q + 1 , and not minimal if 2 ≤ s ≤ 3 ≤ q . In the case where 3 < s ≤ q , for some sunflowers, the corresponding linear codes are minimal, whereas for some other sunflowers, the corresponding linear codes are not minimal. [ABSTRACT FROM AUTHOR]

This review paper aims to examine the use of polar codes in public key cryptosystems, providing a comprehensive overview of the state-of-the-art in this field at present. This study thoroughly searches databases such as IEEE Xplore, Scopus, and the ACM digital library to locate relevant research articles. In this study, we demonstrate the use of polar codes in public key cryptography and provide valuable insights for researchers interested in this field of research. The paper identifies several areas for further research and development, such as improved security and reliability of polar code-based cryptosystems. An analysis of the review highlights the major challenges and open research questions in this area, including the need for efficient key generation algorithms and the trade-offs between security and performance. An analysis of various existing encryption techniques as well as their security proofs is provided in the tables. [ABSTRACT FROM AUTHOR]

Since its inception in the sociological field more than fifty years, Grounded Theory Methodology (GTM) has been extended to a range of research areas but there still be confusions and misconceptions about its history and methodological principles. This paper attempts to provide a full theoretical overview of the history of GTM and allows limiting misconceptions about the methodology itself. Drawing upon social sciences perspectives, the paper begins by briefly shedding the light on two major related concepts: Grounded Theory (GT) and Grounded Theory Methodology (GTM). It continues by defining what GTM really means and outlines the reasons behind The Discovery of grounded theory and further focuses on the big split that gave birth to three schools of Grounded Theory Methodology: Glaserian, Straussian and Constructivist GTM. [ABSTRACT FROM AUTHOR]

A graph G with acyclic coloring has no two adjacent vertices with the same color and no bichromatic cycle. Also, the coloring results in a forest when any two-color classes are combined. The concept of acyclic coloring plays a pivotal role in the computation of Hessians, Kekule structures classification, coding theory, and statistical mechanics. In this paper, the acyclic chromatic number of generalized fan graph, generalized Möbius ladder graph and flower snark graph have been determined. [ABSTRACT FROM AUTHOR]

Proof testing is used to remove weak ceramic components from a manufactured batch in turn truncating the lower end of the strength‐failure probability distribution. With proof testing, components are subjected to stresses comparable to those observed during service load histories resulting in the remaining ones having assured minimum reliabilities. The degree to which the reliability of the remaining parts is assured depends on the proof test protocol, which includes multiple variables such as stress level, realistic sampling of flaws, complex geometries, multiaxial stresses, transient loading histories, time‐dependent damage due to slow crack growth and/or fatigue, and change in material behavior due to temperature and/or environmental conditions. This paper advances the early work done on proof test theory, which was limited to simple stress states, to include general stress and material states. The theory introduced in this paper takes into account practical conditions often encountered in industrial applications. This advanced proof test theory was coded into the CARES/Life code (Ceramics Analysis and Reliability Evaluation of Structures/Life). Examples are provided to demonstrate the theory. [ABSTRACT FROM AUTHOR]

Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing minimal binary codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity, are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension $m$ and length less than or equal to $m(m+1)/2$ are obtained. Besides, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields additionally to symmetric cryptography, such as coding theory and secret sharing schemes. [ABSTRACT FROM AUTHOR]

In its most elementary form, compressed sensing studies the design of decoding algorithms to recover a sufficiently sparse vector or code from a lower dimensional linear measurement vector. Typically it is assumed that the decoder has access to the encoder matrix, which in the combinatorial case is sparse and binary. In this paper we consider the problem of designing a decoder to recover a set of sparse codes from their linear measurements alone, that is without access to encoder matrix. To this end we study the matrix factorisation task of recovering both the encoder and sparse coding matrices from the associated linear measurement matrix. The contribution of this paper is a computationally efficient decoding algorithm, Decoder-Expander Based Factorisation, with strong performance guarantees. Under mild assumptions on the sparse coding matrix and by deploying a novel random encoder matrix, we prove that Decoder-Expander Based Factorisation recovers both the encoder and sparse coding matrix at the optimal measurement rate with high probability and from a near optimal number of measurement vectors. In addition, our experiments demonstrate the efficacy and computational efficiency of our algorithm in practice. Beyond compressed sensing, our results may be of interest for researchers working in areas as diverse as linear sketching, coding theory, matrix compression and dictionary learning. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to explore different types of packaging paradoxes and the reasons for their existence in food supply chains. Design/methodology/approach: The research uses a multiple case study approach with rich empirical data from seven leading companies in Swedish food supply chains. The research uses coding and a paradox theory lens to analyse packaging paradoxes, both within and between companies in a supply chain. Findings: The paper provides a novel theoretical lens which uses comprehensive empirical data to identify and categorise four types of packaging paradoxes on two system levels in food supply chains. It presents detailed descriptions of, and underlying reasons for, the paradoxes. It also discusses strategies required to manage packaging paradoxes. Research limitations/implications: Future research should confirm and extend the findings in this study by incorporating data from companies in other countries. It should cover the importance of paradoxes, their impact on company performance and innovation, and how different paradoxes are related to each other. It should also investigate strategies to manage paradoxes further. Practical implications: The findings should help companies acknowledge and identify management principles for packaging paradoxes in food supply chains. Originality/value: It is the first study which systematically explores packaging paradoxes in food supply chains. The study offers a new approach to understand the complexity of packaging decisions in food supply chains. It contributes to the packaging logistics literature by extending theoretical knowledge about conflicts of interest related to packaging. The management discussion offers initial insights into management of packaging paradoxes and directions for future research. [ABSTRACT FROM AUTHOR]

In order to improve the online control effect of small- and medium-sized electromechanical equipment, this paper studies the online control method of small- and medium-sized electromechanical equipment combined with the deep audit network. Moreover, starting from the signal basis of online control signal recognition and channel coding recognition of electromechanical equipment, this paper determines the type of online control signal and channel coding type of electromechanical equipment required for the experiment. This paper compares the signals under different types of online control signals for electromechanical devices and clarifies the differences in amplitude, frequency, phase, and spectrum in time-domain waveforms. In addition, through the elaboration of different channel coding theories, this paper clarifies the differences in coding methods. Through the experimental research, it can be seen that the online control effect of the small- and medium-sized electromechanical equipment based on the deep neural network proposed in this paper is very good. [ABSTRACT FROM AUTHOR]

Let K = Q (d) with d > 0 square-free and let O K denote the ring of integers of K. Let C ⊂ R n be a linear code where R is F 4 if d ≡ 5 (mod 8) and F 2 × F 2 if d ≡ 1 (mod 8) . One has a surjective ring homomorphism ρ : O K n → R n given by reduction modulo (2 O K) n . The inverse image Λ (C) : = ρ - 1 (C) is a lattice associated to the code C . One can associate to Λ C a theta series Θ Λ d (C) . In this paper we consider how the theta series varies as one varies the value d. In particular, we show that for d , d ′ with d > d ′ , one has Θ Λ d (C) = Θ Λ d ′ (C) + O q d ′ + 1 2 . [ABSTRACT FROM AUTHOR]

APN power functions are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. Let p be an odd prime and n be a positive integer. Let F (x) = x d be a power function over F p n , where d = 3 p n - 1 4 when p n ≡ 3 (mod 8) and d = p n + 1 4 when p n ≡ 7 (mod 8) . When p n > 7 , F is an APN function, which is proved by Helleseth et al. (IEEE Trans Inform Theory 45(2):475–485, 1999). In this paper, we study the differential spectrum of F. By investigating some system of equations, the number of solutions of certain system of equations and consequently the differential spectrum of F can be expressed by quadratic character sums over F p n . By the theory of elliptic curves over finite fields, the differential spectrum of F can be investigated by a given p. It is the fourth infinite family of APN power functions with nontrivial differential spectrum. [ABSTRACT FROM AUTHOR]

Snake graphs are connected planar graphs consisting of a finite sequence of adjacent tiles (squares) T 1 , T 2 , ... , T n . In this case, for 1 ≤ j ≤ n − 1 , two consecutive tiles T j and T j + 1 share exactly one edge, either the edge at the east (west) of T j ( T j + 1 ) or the edge at the north (south) of T j ( T j + 1 ). Finding the number of perfect matchings associated with a given snake graph is one of the most remarkable problems regarding these graphs. It is worth noting that such a number of perfect matchings allows a bijection between the set of snake graphs and the positive continued fractions. Furthermore, perfect matchings of snake graphs have also been used to find closed formulas for cluster variables of some cluster algebras and solutions of the Markov equation, which is a well-known Diophantine equation. Recent results prove that snake graphs give rise to some string modules over some path algebras, connecting snake graph research with the theory of representation of algebras. This paper uses this interaction to define Brauer configuration algebras induced by schemes associated with some multisets called polygons. Such schemes are named Brauer configurations. In this work, polygons are given by some admissible words, which, after appropriate transformations, permit us to define sets of binary trees called groves. Admissible words generate codes whose energy values are given by snake graphs. Such energy values can be estimated by using Catalan numbers. We include in this paper Python routines to compute admissible words (i.e., codewords), energy values of the generated codes, Catalan numbers and dimensions of the obtained Brauer configuration algebras. [ABSTRACT FROM AUTHOR]

Erasure codes are essential for providing efficient resilience against node failures in distributed storage. Typically, an $[n, k]$ erasure code encodes $k$ symbols into $n$ symbols which are then stored in different nodes. Recent work by Kadekodi et al. shows that the failure rates of storage nodes vary significantly over time, and that changing the rate of the code (via a change in $n$ and $k$) in response to such variations provides substantial storage space savings. However, the resource overhead of re-encoding the already encoded data is prohibitively high. We present a new theoretical framework formalizing code conversion—the process of converting data encoded with an $[n^{ I}, k^{ I}]$ code into data encoded with an $[{n^{ F}}, {k^{ F}}]$ code while maintaining desired decodability properties. We then introduce convertible codes, a new class of codes that allow for code conversions in a resource-efficient manner. This paper begins the study on convertible codes by focusing on linear MDS codes and the access cost of conversion. We derive a lower bound on the access cost of conversion and present an explicit optimal construction matching this bound for an important subclass of conversions. Additionally, we propose constructions with low field-size requirement for a broad subset of parameters. Our results show that it is possible to achieve code conversions with significantly less resources than the default approach of re-encoding for a wide range of parameters. [ABSTRACT FROM AUTHOR]

Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$ -ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$ -ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes. [ABSTRACT FROM AUTHOR]

Linear codes with few weights have applications in consumer electronics, communications, data storage systems, secret sharing, authentication codes, and association schemes. As a special class of linear codes, minimal linear codes have important applications in secret sharing and secure computation of data between two parties. The construction of minimal linear codes with new and desirable parameters is an interesting research topic in coding theory and cryptography. Recently, Mesnager et.al. stated that “constructing linear codes with good parameters from non-weakly regular bent functions is an interesting problem.” The goal of this paper is to construct linear codes with two or three weights from non-weakly regular bent functions over finite fields and analyze the minimality of the constructed linear codes. In doing so, we draw inspiration from a paper by Mesnager in which she constructed linear codes with small weights from weakly regular bent functions based on a generic construction method. First, we recall the definitions of the subsets $B_{+}(f)$ and $B_{-}(f)$ associated with a non-weakly regular bent function $f$. Next, we construct two- or three-weight linear $p$ -ary codes on these sets using duals of the non-weakly regular bent functions that are also bent. We note that the constructed linear codes are minimal in almost all cases. Moreover, when $f$ is a non-weakly regular bent function in a certain subclass of Generalized Maiorana-McFarland bent functions, we determine the weight distributions of the corresponding linear codes. As far as we know, the construction of linear codes from non-weakly regular bent functions over finite fields is first studied in the literature by the second author in his dissertation. [ABSTRACT FROM AUTHOR]

Background: Normalisation Process Theory (NPT) is frequently used to inform qualitative research that aims to explain and evaluate processes that shape late-stage translation of innovations in the organisation and delivery of healthcare. A coding manual for qualitative researchers using NPT will facilitate transparent data analysis processes and will also reduce the cognitive and practical burden on researchers.Objectives: (a) To simplify the theory for the user. (b) To describe the purposes, methods of development, and potential application of a coding manual that translates normalisation process theory (NPT) into an easily usable framework for qualitative analysis. (c) To present an NPT coding manual that is ready for use.Method: Qualitative content analysis of papers and chapters that developed normalisation process theory, selection and structuring of theory constructs, and testing constructs against interview data and published empirical studies using NPT.Results: A coding manual for NPT was developed. It consists of 12 primary NPT constructs and conforms to the Context-Mechanism-Outcome configuration of realist evaluation studies. Contexts are defined as settings in which implementation work is done, in which strategic intentions, adaptive execution, negotiating capability, and reframing organisational logics are enacted. Mechanisms are defined as the work that people do when they participate in implementation processes and include coherence-building, cognitive participation, collective action, and reflexive monitoring. Outcomes are defined as effects that make visible how things change as implementation processes proceed and include intervention mobilisation, normative restructuring, relational restructuring, and sustainment.Conclusion: The coding manual is ready to use and performs three important tasks. It consolidates several iterations of theory development, makes the application of NPT simpler for the user, and links NPT constructs to realist evaluation methods. The coding manual forms the core of a translational framework for implementation research and evaluation. [ABSTRACT FROM AUTHOR]

Orthogonal array is one of the important research subjects in combinatorial design theory and experimental design theory, and it is widely applied to statistics, computer science, coding theory and cryptography. There are many constructions and results for orthogonal array of strength 2, however the results on orthogonal array of strength t ≥ 3 are rare. In this paper, we first present two new effective constructions for orthogonal arrays of strength t ≥ 3 based on large sets of orthogonal arrays. Second, many infinite families of large sets of orthogonal arrays are obtained and then some new series of orthogonal array of strength t ≥ 3 are produced. [ABSTRACT FROM AUTHOR]

In the platform economy, platform enterprises connect producers and customers, matching diverse supply with diverse demand to achieve profit goals. The big data assets and marketing relationships owned by platform enterprises are complementary assets, and their modular organizational structure is crucial for enhancing enterprise value innovation. The existing literature lacks in-depth exploration of the impact of complementary assets on platform enterprises. Therefore, the research purpose of this article is to investigate the impact of complementary assets and organizational modularity on platform enterprise value innovation. Taking Meituan as the research object, this paper adopts the single case-study method of grounded theory coding to explore the relationship between complementary assets, organizational modularity, and platform enterprise value innovation. Research has shown that organizational modularity includes two dimensions, dependency and structural arrangement, while value innovation includes three dimensions, customer value innovation, product value, and innovation ability. Among them, complementary assets and organizational modularity both have a positive impact on platform enterprise value innovation, and organizational modularity plays a moderating role in the impact of complementary assets on platform enterprise value innovation. The research results have enriched the research on complementary assets, organizational modularity, and value innovation theory, providing a theoretical basis and inspiration for enterprise organizational design and better utilization of complementary assets. [ABSTRACT FROM AUTHOR]

Boosted by cryptography and coding theory applications and rich connections to objects from geometry and combinatorics, bent functions and related functions developed into a lively research area. In the mid-seventies, Rothaus initially introduced bent functions in the Boolean case, but later, they extended in the p-ary case where p is any prime. Such an extension brought more rich connections to bent functions. In addition to the theory of Fourier transform, handling p-ary bent functions (where p is an odd prime), analyzing their properties and designing them require using the theory of cyclotomic fields. Such a family is classified into weakly regular bent and regular bent. Compared to the Boolean case, the class of p-ary bent functions inherits a much larger variety of properties. For example, it contains the class of dual-bent functions as a proper subclass, which again includes weakly regular bent functions as a proper subclass. Weakly regular bent functions can also be employed in many domains. In particular, they have been widely used in designing good linear codes for several applications (such as secret sharing and two-party computation), association schemes, and strongly regular graphs. More exploration is still needed despite a lot of interest and attention in this topic to increase our knowledge of bent functions and design them. In particular, only a few constructions of weakly regular bent functions have been presented in the literature. Nice construction methods of these functions have been given by Tang et al (IEEE Trans Inf Theory 62(3):1166–1176, 2016). All the known infinite families of weakly regular bent functions belong to a class (denoted by Tang et al. in the previous reference) R F . This paper is devoted to weakly regular bent functions, and its objective is twofold. First, it aims to generate new infinite families of weakly regular bent functions living outside R F and secondly, to exploit the constructed functions to design new families of p-ary linear codes and investigate their use for some standard application after studying this minimality based on their weight distributions. More specifically, we present several classes of weakly regular bent functions obtained from monomial bent functions by modifying the values of some known weakly regular bent functions on some subsets of the finite field F p n (n is a positive integer). Our weakly regular bent functions are of degree either p or p + 1 2 . We also explicitly determine their corresponding dual functions. Finally, we exploit our constructions to derive four new classes of linear codes with three to seven weights. We also show that two of them lead to minimal codes, which are more appropriate for several concrete applications. [ABSTRACT FROM AUTHOR]

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In this paper, we investigate a modification of the method of data generation for multiple code paths within a single launch of the genetic algorithm. This method allows the consideration of the remoteness of paths initiated by different test cases by introducing an additional additive component into the fitness function. Previous studies have shown that the parameter defining the relationship between the different components of the fitness function has a rather strong effect on code coverage. To eliminate this effect, we propose the modification of the first component of the fitness function, which is responsible for path complexity. This modification is based on a dynamic change in code statement weights between generations to achieve greater population diversity. We propose several methods for implementing this modification, divided into two groups. In the first group, the statement weights change depending only on the fact of statement coverage in a generation, and the rate of change depends on the number of previous generations in which it was covered. In the second group, the rate of change depends on the proportion of statement coverage by the test sets in the previous generation. Each of the proposed methods is investigated to achieve complete coverage with different values of the parameter defining the ratio of the components of the fitness function. As a result, the best method is determined, which eliminates the need to determine this parameter for each testing code, thus achieving a greater universality for the algorithm. [ABSTRACT FROM AUTHOR]

Formulating the research question is a key but complex task in qualitative studies. The question should be framed consistently with the approach chosen; in other words, question and approach are interdependent. This article aims to contribute to the understanding of the nature of the research question; to this end, we address its meaning by presenting two qualitative approaches: applied hermeneutic phenomenology and grounded theory. Although both approaches draw from the experience of the participants in research, they are vastly different and comparing them provides an exemplar of the important decision-making required for crafting research questions epistemologically aligned with their designs. From this starting point, in this paper we discuss the specific nature of the research question in two qualitative approaches: applied hermeneutic phenomenology and grounded theory. In hermeneutic phenomenology, the question is related to the phenomena of lived experience, with the goal to describe its disclosed/apprehended essence, in the meaning structures of the studied experience. In grounded theory, the question is driven by the purpose of developing theory in a specific field, either to grant exhaustive knowledge in the exploration of hitherto little-known situations or to further our understanding of human behavior. Taking the research question as our central theme, we divide this article into three parts. First, we specify the main characteristics of each of the two approaches. Second, we discuss the formulation of the research question in each. Third, we offer an example of a research question for each of the approaches. [ABSTRACT FROM AUTHOR]

The present Special Issue of Designs, Codes and Cryptography is dedicated to Aart Blokhuis, one of the leading mathematicians in finite geometry and combinatorics. References 1 Ball S, Blokhuis A, Mazzocca F. Maximal arcs in Desarguesian planes of odd order do not exist. The third editor of this special issue, Michel Lavrauw, was a PhD student of Aart between 1997 and 2001. Another challenging problem at the time was to prove the non-existence of maximal arcs in Galois planes of odd order; this was achieved by Aart and his coauthors Simeon Ball and Francesco Mazzocca in [[1]]. [Extracted from the article]

This paper investigates the theory of sum-rank-metric codes for which the individual matrix blocks may have different sizes. Various bounds on the cardinality of a code are derived, along with their asymptotic extensions. The duality theory of sum-rank-metric codes is also explored, showing that MSRD codes (the sum-rank analogue of MDS codes) dualize to MSRD codes only if all matrix blocks have the same number of columns. In the latter case, duality considerations lead to an upper bound on the number of blocks for MSRD codes. The paper also contains various constructions of sum-rank-metric codes for variable block sizes, illustrating the possible behaviours of these objects with respect to bounds, existence, and duality properties. [ABSTRACT FROM AUTHOR]

Krawtchouk polynomials (KPs) are discrete orthogonal polynomials associated with the Gauss hypergeometric functions. These polynomials and their generated moments in 1D or 2D formats play an important role in information and coding theories, signal and image processing tools, image watermarking, and pattern recognition. In this paper, we introduce a new four-term recurrence relation to compute KPs compared to their ordinary recursions (three-term) and analyse the proposed algorithm speed. Moreover, we use Clenshaw's technique to accelerate the computation procedure of the Krawtchouk moments (KMs) using a fast digital filter structure to generate a lattice network for KPs calculation. The proposed method confirms the stability of KPs computation for higher orders and their signal reconstruction capabilities as well. The results show that the KMs calculation using the proposed combined method based on a four-term recursion and Clenshaw's technique is reliable and fast compared to the existing recursions and fast KMs algorithms. [ABSTRACT FROM AUTHOR]

Audio data compression is used to reduce the transmission bandwidth and storage requirements of audio data. It is the second stage in the audio mastering process with audio equalization being the first stage. Compression algorithms such as BSAC, MP3 and AAC are used as standards in this paper. The challenge faced in audio compression is compressing the signal at low bit rates. The previous algorithms which work well at low bit rates cannot be dominant at higher bit rates and vice-versa. This paper proposes an altered form of vector quantization algorithm which produces a scalable bit stream which has a number of fine layers of audio fidelity. This modified form of the vector quantization algorithm is used to generate a perceptually audio coder which is scalable and uses the quantization and encoding stages which are responsible for the psychoacoustic and arithmetical terminations that are actually detached as practically all the data detached during the prediction phases at the encoder side is supplemented towards the audio signal at decoder stage. Therefore, clearly the quantization phase which is modified to produce a bit stream which is scalable. This modified algorithm works well at both lower and higher bit rates. Subjective evaluations were done by audio professionals using the MUSHRA test and the mean normalized scores at various bit rates was noted and compared with the previous algorithms. [ABSTRACT FROM AUTHOR]

Blockchain, with its characteristics of non-tamperability and decentralization, has had a profound impact on various fields of society and has set off a boom in the research and application of blockchain technology. However, blockchain technology faces the problem of data availability attacks during its application, which greatly limits the scope and domain of blockchain applications. One of the most advantageous researches to address this problem is the scalable data availability solution that integrates coding theory design into the Merkle tree promise. Based on this scheme, this paper combines a zero-knowledge accumulator with higher efficiency and security with local repair coding, and proposes a data availability scheme with strong dataset privacy protection. The scheme first encodes the data block information on the blockchain to ensure tamper-proof data, and then uses a zero-knowledge accumulator to store the encoded data block information. Its main purpose is to use zero-knowledge property to protect the accumulation set information stored in the accumulator from being leaked and to ensure that no other information about the accumulation set is revealed during the data transmission. It fundamentally reduces the possibility of attackers generating fraudulent information by imitating block data and further resists data availability attacks. [ABSTRACT FROM AUTHOR]