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A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of non-scalar multiplications required to evaluate it. This quantity is also described as its tensor rank, which is the smallest number of rank-1 matrices whose span contains its first slice space. In this paper we derive upper bounds on the tensor ranks of certain classes of 3-tensors and give explicit constructions of sets of rank-1 matrices containing their first slice spaces. We also show how these results can be applied in coding theory to derive upper bounds on the tensor rank of some rank-metric codes. In particular, we compute the tensor rank of some families of F q m -linear codes and we show that they are extremal with respect to Kruskal's tensor rank bound. [ABSTRACT FROM AUTHOR]

Space-filling designs are useful for exploring the relationship between the response and factors, especially when the true model is unknown. The wrap-around L 2 -discrepancy is an important measure of the uniformity, and has often been used as a type of space-filling criterion. However, most obtained designs are generated through stochastic optimization algorithms, and cannot achieve the lower bound of the discrepancies and are only nearly uniform. Then deterministic construction methods for uniform designs are desired. This paper constructs uniform designs under the wrap-around L 2 -discrepancy by generator matrices of linear codes. Several requirements on the generator matrices, such as a necessary and sufficient condition for generating uniform designs, are derived. Based on these, two simple deterministic constructions for uniform designs are given. Some examples illustrate the effectiveness of them. Moreover, the resulting designs can be regarded as a generalization of good lattice point sets, and also enjoy good orthogonality. • The deterministic construction methods for uniform designs under the wrap-around L 2 -discrepancy are given. • A necessary and sufficient condition for generating uniform designs are derived. • The resulting designs can be regarded as a generalization of good lattice point sets, and also enjoy good orthogonality. [ABSTRACT FROM AUTHOR]

There are many infinite families of 2-designs and 3-designs supported by linear codes in the literature. Recently, several infinite families of 4-designs from linear codes were discovered. But no infinite family of linear codes supporting an infinite family of nontrivial simple 5-designs is reported in the literature. It is not easy to construct 5-designs from sporadic codes. Only a small number of sporadic 5-designs supported by linear codes are available in the literature. In this paper, we construct eight 5-designs from some lifted codes of some known linear codes. • There are many infinite families of 2-designs and 3-designs supported by linear codes in the literature. • Recently, several infinite families of 4-designs from linear codes were discovered. • But no infinite family of linear codes supporting an infinite family of nontrivial simple 5-designs is reported in the literature. • It is not easy to construct 5-designs from sporadic codes. Only a small number of sporadic 5-designs supported by linear codes are available in the literature. • In this paper, we construct eight new 5-designs from some lifted codes of some known linear codes. [ABSTRACT FROM AUTHOR]

• Efficient nonlinear compression codes for circuit testing. • Fully deterministic code creation methodology. • Easy way how to get large codes with no computation time increase. The nonlinear block codes have significantly better testing abilities than the linear ones used in Built-In Self Test Equipment or test pattern decompressors. Unfortunately, no sufficiently effective deterministic code creation strategy has been discovered till now. This paper deals with a deterministic nonlinear code search strategy and a strategy based on extending linear compression codes by nonlinear check bits that improve the usability of decompressed patterns for testing circuits with more inputs. The earlier works used a purely or partially random search of the nonlinear check-bits truth tables to construct the first nonlinear structures. Here, we derive deterministic rules that can be used for the iterative process of code generation with no necessity of any random search for codes with the number of specified bits equal to three. We compare the code parameters of codes based on linear code extension, direct deterministic code search methods, and previously published results. The most extensive codes still need to be constructed with a partial random search strategy, but keeping the restrictions introduced in the paper causes the search to be substantially more efficient. [ABSTRACT FROM AUTHOR]

In this paper we focus on modules over a finite chain ring R of size q s. We compute the density of free modules of R n , where we separately treat the asymptotics in n , q and s. In particular, we focus on two cases: one where we fix the length of the module and one where we fix the rank of the module. In both cases, the density results can be bounded by the Andrews-Gordon identities. We also study the asymptotic behaviour of modules generated by random matrices over R. Since linear codes over R are submodules of R n we get direct implications for coding theory. For example, we show that random codes achieve the Gilbert-Varshamov bound with high probability. [ABSTRACT FROM AUTHOR]

Constant dimension codes, a specialized subclass of subspace codes, have garnered significant attention in recent years due to their applications in random network coding. Among these codes, large cyclic constant dimension codes, endowed with optimal minimum distance, offer efficient solutions for encoding and decoding algorithms. This paper is dedicated to the design of cyclic constant dimension codes that achieve the dual goals: maximizing code size while maintaining optimal minimum distance. We first explore a new form of Sidon spaces and employ this new form to construct several families of cyclic subspace codes. Our codes have optimal minimum distances and have more codewords than previous constructions in the literature. [ABSTRACT FROM AUTHOR]

The Singleton-type bound that offers an upper limit on the minimum Hamming distance for locally repairable codes (LRCs) has been studied by a lot of literatures. However, the work to determine tight Singleton-type bound for LRCs under all parameters is still incomplete. This paper focuses on the tightness of Singleton-type bound for general LRCs (linear and nonlinear). By exploring how the structure of repair sets affects the minimum Hamming distance, some properties for minimum Hamming distance of LRCs are derived. Through these properties, the following results are obtained: 1. The necessary conditions for LRCs whose minimum Hamming distance reaches the previous Singleton-type bound are provided. Due to these necessary conditions, the previous Singleton-type bound is proved to be unreachable under a number of parameters, and the tight Singleton-type bounds are given under some of these parameters. 2. Under a class of parameters where the previous Singleton-type bound for linear LRCs has been proved unreachable in the literatures, we improve Singleton-type bound and show that the new Singleton-type bound is tight. In addition, based on the existing LRC constructions, the previous Singleton-type bound for LRCs is shown to be tight under some other parameters. Finally, for an importance case of LRCs (i.e., every repair set is a 1-erasure correcting code), the Singleton-type bound under three classes of parameters is further improved. [ABSTRACT FROM AUTHOR]

Although phase field model has demonstrated excellent performance in dealing with complex fracture, its development and application are hindered by computational costs. In order to reduce the computational time, a novel method is proposed in this paper that couples the phase field model with the classical linear elastic theory model. The solution domain is divided into different subdomains, and corresponding solution scheme is used for each region. The phase field model is solved only in the subdomain where the crack is about to propagate to determine the crack path, while the rest of the solution domain is solved using a linear elastic theory model, allowing for coarser discretization to improve computational efficiency and reduce computational costs. A transition subdomain is included between the two models to ensure that the results from the different models are harmonized at the boundary and to improve convergence. The coupling model is implemented in ABAQUS using user defined element and built-in features, and the model is applied to both static and dynamic fracture analysis. The coupling model is validated through representative examples, and the results show that the presented model can not only effectively portray the static and dynamic fracture processes and accurately describe the crack propagation, but also reduce the computational costs. • A novel method for coupling phase field model with classical linear elastic theory model is proposed. • The coupling model is implemented in ABAQUS using user defined element and built-in features. • The coupling model is applied to static and dynamic fracture analysis. • The presented strategy can effectively reduce computational costs. [ABSTRACT FROM AUTHOR]

Abstract As a development of sparse coding, while retaining the advantage of sparse coding in classification, Locality-constrained Linear Coding(LLC) greatly improves the time efficiency of appearance modeling. However, in order to further promote the performance of real-time and develop a tracking algorithm that can be applied to both visible light images and infrared images, this paper proposes a tracking algorithm using LLC and saliency map under the framework of particle filtering. It is universally acknowledged that number of particles determines the accuracy of tracker under the framework of particle filtering. Unfortunately, the increase in the number of particles leads to the augment of computational burden. Therefore, the basic idea of the proposed algorithm is to reduce the computational number of observation vectors while keeping the effective number of particles and achieve the goal of strengthening the real-time performance of tracker. The proposed algorithm firstly uses spectral residual to obtain a saliency map of the current frame and then computes the saliency score of each particle. Secondly, several particles are eliminated directly according to the difference between the saliency score of the particle in the current frame and the target score in the previous frame. Thirdly, LLC is used to compute the observation vector for the rest particles and complete tracking tasks. Both quantitative and qualitative experimental results demonstrate that the proposed algorithm performs favorably against the nine state-of-the-art trackers on twelve challenging test sequences including six visible light sequences and six infrared sequences. In addition, related experimental results reveal that the proposed algorithm decreases the computational complexity and has the better tracking performance compared with the tracker just using LLC in the framework of particle filtering. Highlights • This paper proposes a tracking algorithm combining LLC and spectral residual. Because of the low computation cost of spectral residual-based saliency analysis, the proposed algorithm reduces the computational complexity. • Under the framework of particle filter, the new update mechanism of particle weight are designed according to the saliency score. • The proposed algorithm can be used for target tracking in both visible light and infrared images. [ABSTRACT FROM AUTHOR]

Linear codes over finite chain rings meeting the Griesmer bound are particularly interesting since they are closely related to minihyper in projective Hjelmslev spaces. In this paper, motivated by the nice article (Li et al., 2018 [23]), two classes of linear codes with few non-zero Hamming weights over the finite chain ring R = F q + u F q are constructed, and one of them meets the Griesmer bound over finite commutative local rings. By a Gray map, we obtain that the Gray images of the constructed codes over the ring R are linear codes with two weights over the finite field F q with q elements, and these codes are optimal with respect to the well-known Griesmer bound under certain conditions. Our results give an affirmative answer to two (related) open questions raised in the article mentioned above. Remarkably, some of the constructed linear codes over R presented in this paper correspond to minihypers in projective Hjelmslev spaces over finite chain rings. Besides, their Gray images can be used to construct secret sharing schemes with nice access structures and other applications thanks to the minimality property that possesses our constructed codes. [ABSTRACT FROM AUTHOR]

Recently, linear codes with few weights have been widely studied, since they have applications in data storage systems, communication systems and consumer electronics. In this paper, we present a class of three-weight and five-weight linear codes over F p , where p is an odd prime and F p denotes a finite field with p elements. The weight distributions of the linear codes constructed in this paper are also settled. Moreover, the linear codes illustrated in the paper may have applications in secret sharing schemes. [ABSTRACT FROM AUTHOR]

This paper describes a test response compaction method that preserves diagnostic information and enables performing a test-per-clock offline test. The test response compaction system is based on a chain of T flip-flops. The T flip-flop signature chain can preserve the information about the positions of the erroneous test response occurrence and the information about the clock cycle when the erroneous test responses occurred. This information can be used for diagnostic purposes. An algorithm that localizes errors according to the T flip-flop chain output is presented. The paper discusses the possible benefits and limitations of the proposed test pattern compaction scheme. The influence of multiple errors on detection and localization capability of the compaction system and hardware overhead is discussed in the paper as well. The probability of error masking is analyzed, the proposed scheme provides substantially lower masking probability than a D flip-flop chain and a MISR. The scheme can spare the test time by the test-per-clock arrangement. The hardware overhead and reached test time are given for several benchmark circuits in the paper as well. [ABSTRACT FROM AUTHOR]

This paper constructs a class of lattices whose discrete analogues are variable-length non-linear codes. The well-known discrete analogue of lattices and linear codes inspires our approach. We next design a variable length binary non-linear code called automorphism orbit code from a finite abelian p -group of rank greater than 1, where p is a prime. For each finite abelian p -group, codewords of the automorphism orbit code are variable-length codewords called automorphism orbit codewords. The homomorphisms between groups determine homomorphism codes , whereas automorphism orbit codes are specified by partitions of a number, orbits of group action, homomorphisms and automorphisms of groups. For some groups G and H , we shall use elements of H o m (G , H) to create a cover relation for bit strings of codewords of an automorphism orbit code and formulate a lattice of variable length non-linear codes. [ABSTRACT FROM AUTHOR]

Let q be a prime power and m a positive integer. Suppose that a ≥ 2 is a factor of q m − 1 such that m is the multiplicative order of q modulo n : = q m − 1 a. Firstly, for m odd and m even, we present some necessary and sufficient conditions on dual-containing non-primitive BCH codes of length n with the maximum designed distances over F q , respectively. The results show that the maximum designed distances of dual-containing BCH codes in this paper are larger than those in [2, Theorem 3]. Secondly, we explicitly determine the dimensions of some dual-containing non-primitive BCH codes of length n , and finally, we construct some new quantum codes with relatively large minimum distances. [ABSTRACT FROM AUTHOR]

Generalizing the linear complementary duals, the linear complementary pairs and the hull of codes, we introduce the concept of ℓ -dimension linear intersection pairs (ℓ -DLIPs) of codes over a finite commutative ring (R), for some positive integer ℓ. In this paper, we study ℓ -DLIP of codes over R in a very general setting by a uniform method. Besides, we provide a necessary and sufficient condition for the existence of a non-free (or free) ℓ -DLIP of codes over a finite commutative Frobenius ring. In addition, we obtain a generator set of the intersection of two constacyclic codes over a finite chain ring which helps us to get an important characterization of ℓ -DLIP of constacyclic codes. Finally, the ℓ -DLIP of constacyclic codes over finite chain rings are used to construct new entanglement-assisted quantum error correcting (EAQEC) codes. [ABSTRACT FROM AUTHOR]

The fifth generation (5G) wireless communication standard aims to integrate multiple networks, offering high throughput, large bandwidth, and ultra-low latency. Importantly, channel coding is now a crucial component of 5G wireless communication systems. Polar codes, introduced by Erdal Arikan, are linear block codes chosen for 5G control channels due to their capacity-reaching properties and efficient encoding/decoding algorithms. Polar codes are expected to be vital for achieving high reliability and low latency in 6G wireless communication systems, with current research focused on enhancing polar decoders. Two main polar decoder types exist: successive cancellation (SC) with lower complexity but limited throughput, and belief propagation (BP) with higher throughput but increased complexity. Addressing these challenges, deep learning (DL) techniques enhance both SC and BP decoders, leveraging fault tolerance, parallel and distributed processing ability, scalability, and better self-learning capability. The application of the DL technique in polar decoders lowers the cost of computation and significantly enhances the decoders' performance. This paper surveys novel DL-based SC/BP polar decoder approaches, evaluating error performance, decoding competence, learning efficiency, memory requirements, and hardware complexity. The review concludes that DL-based polar decoders hold great potential for enhancing performance in 5G networks and beyond. [ABSTRACT FROM AUTHOR]

Based on good algebraic structures and practicabilities, generalized quasi-cyclic (GQC) codes play important roles in coding theory. In this paper, we study some results on GQC codes over Z 4 including normalized generating sets, minimal generating sets and normalized generating sets of their dual codes. As an application, new Z 4 -linear codes and good binary nonlinear codes are constructed from GQC codes over Z 4. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some new polynomials associated to linear codes over F q. In particular, we introduce the notion of split complete Jacobi polynomials attached to multiple sets of coordinate places of a linear code over F q , and give the MacWilliams type identity for it. We also give the notion of generalized q -colored t -designs. As an application of the generalized q -colored t -designs, we derive a formula that obtains the split complete Jacobi polynomials of a linear code over F q. Moreover, we define the concept of colored packing (resp. covering) designs. Finally, we give some coding theoretical applications of the colored designs for Type III and Type IV codes. [ABSTRACT FROM AUTHOR]

In this paper we classify 2 -designs and 3 -designs with 28 or 36 points admitting a transitive action of the unitary group U ( 3 , 3 ) on points and blocks. We also construct 2 -designs and 3 -designs with 56 or 63 points and strongly regular graphs on 36, 63 or 126 vertices having U ( 3 , 3 ) as a transitive automorphism group. Further, we show that this completes the classification of 3 -designs admitting a transitive action of the group U ( 3 , 3 ) , in terms of parameters. A number of the 3 -designs and 2 -designs obtained in this paper have not been known before up to our best knowledge. [ABSTRACT FROM AUTHOR]

A group-characterizable (GC) random variable is induced by a finite group, called main group, and a collection of its subgroups. The notion extends directly to secret sharing schemes (SSSs). It is known that linear and abelian SSSs can be equivalently described in terms of GC SSSs. In this paper, we present a necessary and sufficient condition for a SSS to be equivalent to a GC one. Using this result, we show that homomorphic SSSs (HSSSs) are equivalent to GC SSSs whose subgroups are normal in the main group. We also present two applications for this equivalent description of HSSSs. One concerns lower bounding the information ratio of access structures for the class of HSSSs, and the other is about the coincidence between statistical, almost-perfect and perfect security notions for the same class. [ABSTRACT FROM AUTHOR]

Recently, linear codes constructed from defining sets have been studied extensively. They may have a few weights if the defining set is chosen properly. Let m and t be positive integers. For an odd prime p , let r = p m and Tr be the absolute trace function from F r to F p . In this paper, for b ∈ F p ∗ , we define D b = ( x 1 , … , x t ) ∈ F r t : Tr ( x 1 + ⋯ + x t ) = b , and determine the complete weight enumerator of a class of p -ary linear codes given by C D b = { c ( a 1 , a 2 , … , a t ) : a 1 , … , a t ∈ F r } , where c ( a 1 , a 2 , … , a t ) = ( Tr ( a 1 x 1 2 + ⋯ + a t x t 2 ) ) ( x 1 , … , x t ) ∈ D b . Then we get their weight enumerators explicitly, which will give us several linear codes with a few weights. As a generalization of Wang et al. (arXiv:1512.03866), this paper extends the result of Ahn et al. (2016). [ABSTRACT FROM AUTHOR]

Galois inner product is a generalization of the Euclidean inner product and Hermitian inner product. The theory on linear codes under Galois inner product can be applied in the constructions of MDS codes and quantum error-correcting codes. In this paper, we construct Galois self-dual codes and MDS Galois self-dual codes from extensions of constacyclic codes. First, we explicitly determine all the Type II splittings leading to all the Type II duadic constacyclic codes in two cases. Second, we propose methods to extend two classes of constacyclic codes to obtain Galois self-dual codes, and we also provide existence conditions of Galois self-dual codes which are extensions of constacyclic codes. Finally, we construct some (almost) MDS Galois self-dual codes using the above results. Some Galois self-dual codes and (almost) MDS Galois self-dual codes obtained in this paper turn out to be new. [ABSTRACT FROM AUTHOR]

In this paper, we mainly determine the weight distributions of three classes of linear codes. Firstly, we prove that two classes of ternary linear codes from the following two planar functions have two or three weights: f ( x ) = x 3 k + 1 2 , x ∈ F 3 m , where k , m are odd, gcd ( m , k ) = 1 , and f ( x ) = x 3 k + 1 , x ∈ F 3 m , where m gcd ( m , k ) is odd. They are exactly a part of the open problem in Ding and Ding (2015 Section IV). Secondly, we construct a new class of binary linear codes with three weights. In particular, the linear codes in this paper have applications in consumer electronics, communication and secret sharing schemes. [ABSTRACT FROM AUTHOR]

In this paper we address the problem of list decoding of linear codes over an integer residue ring Z q , where q is a power of a prime p. The proposed procedure exploits a particular matrix representation of the linear code over Z p r , called the standard form, and the p -adic expansion of the to-be-decoded vector. In particular, we focus on the erasure channel in which the location of the errors is known. This problem then boils down to solving a system of linear equations with coefficients in Z p r . From the parity-check matrix representations of the code we recursively select certain equations that a codeword must satisfy and have coefficients only in the field p r − 1 Z p r . This yields a step by step procedure obtaining a list of the closest codewords to a given received vector with some of its coordinates erased. We show that such an algorithm actually computes all possible erased coordinates, that is, the provided list is minimal. [ABSTRACT FROM AUTHOR]

Galois hulls are generalizations of Euclidean and Hermitian hulls, which have attracted interest because of their important applications in determining the complexity of some algorithms about linear codes and in constructing entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, the whole contribution is two-folded. On one hand, we propose explicit methods to construct linear codes of larger lengths with Galois hulls of arbitrary dimensions from given Galois self-orthogonal codes. Conditions required in our approach are proven to be relatively weak. On the other hand, we apply these results to construct EAQECCs. Two bounds for EAQECCs constructed from linear codes with prescribed dimensional Galois hull are given and EAQECCs with rates greater than or equal to 1 2 and positive net rates can be obtained. We also present many interesting examples to explain visually how these two aspects work. [ABSTRACT FROM AUTHOR]

This paper provides the Generalized Mattson Solomon polynomial for repeated-root polycyclic codes over local rings that gives an explicit decomposition of them in terms of idempotents. It also states some structural properties of repeated-root polycyclic codes over finite fields in terms of matrix product codes. Both approaches provide a description of the ⊥ 0 -dual code for a given polycyclic code. [ABSTRACT FROM AUTHOR]

Recently, the research of trace codes over finite fields has received a lot of attention because some of these codes have good parameters. In this paper, we construct several classes of trace codes from some special classes of linear codes and give their weight distributions, showing that they contain some optimal and almost optimal codes. Furthermore, we determine the parameters of the duals of the trace codes, and show that they are distance-optimal or nearly distance-optimal with respect to the sphere-packing bound. [ABSTRACT FROM AUTHOR]

In this paper, we first study the (non)-degeneracy of a class of quadratic forms. Then with a proper defining set determined by the given quadratic form, we construct a class of linear codes, decide their weight enumerators and show that they are two- or three-weight codes by using the Weil sums. Some of the punctured codes of the discussed linear codes are optimal or distance optimal to the Griesmer bound. [ABSTRACT FROM AUTHOR]

If a linear code is contained in its dual, then it is said to be self-orthogonal. Self-orthogonal codes are an important subclass of linear codes which have many nice applications. In this paper, we mainly study the augmented codes of the linear codes constructed by Hu et al. (2022) [21] , and the subcodes of the codes punctured from the irreducible cyclic codes constructed by Ding and Yang (2013) [11]. The parameters and weight distributions of these codes are explicitly determined. These codes are proved to be self-orthogonal under certain conditions. As an application, these self-orthogonal codes are used to derive optimal or almost optimal quantum codes with new parameters. [ABSTRACT FROM AUTHOR]

This paper concerns the structure and the properties of a special class of combinatorial systems called minimal linear grammars. The role of unambiguous minimal linear grammars is investigated in the framework of the information transmission and coding problem and some related issues. [ABSTRACT FROM AUTHOR]

Let p be an odd prime number. In this paper, we construct 2 (2 p − 3) classes of codes over the ring R = F p + u F p , u 2 = 0 , which are associated with down sets. We compute the Lee weight distributions of the 2 (2 p − 3) classes of codes when the down sets are generated by a single maximal element. Moreover, by using the Gray map of the linear codes over R , we find out 2 (p − 1) classes of p -ary distance optimal linear codes. Two of them meet the Griesmer bound. [ABSTRACT FROM AUTHOR]

Quantum maximal-distance-separable (MDS) codes are an important class of quantum codes. Recently, many scholars utilize constacyclic codes to construct some quantum MDS codes. In this paper, several new families of optimal asymmetric quantum codes and optimal quantum convolutional codes are constructed by using constacyclic codes. Moreover, these quantum codes constructed in this paper are different from the ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper an effective graph learning method is proposed for clustering based on adaptive graph regularizations. Many graph learning methods focus on optimizing a global constraint on sparsity, low-rankness or weighted pair-wise distances, but they often fail to consider local connectivities. We demonstrate the importance of locality by generalizing the Locality-constrained Linear Coding (LLC) for unsupervised learning. Each data sample is expressed as a representation of its nearest neighbors, which naturally leads to a combination of distance regularized features and a Locally Linear Embedding (LLE) decomposition. The representation enforces a locally sparse connection on the data graph that exhibits high discrimination power and is easy to optimize. To improve the learned graph structure and incorporate cluster information, a rank constraint is further imposed on the Laplacian matrix of the data graph so that the connected components match the class number. The obtained representations are smoothed via manifold regularizations on a predefined graph which serves as a prior for graph learning. Finally, we utilize unsupervised Extreme Learning Machine (US-ELM) to learn a flexible and discriminative data embedding. Extensive evaluations show that the proposed algorithm outperforms graph learning counterpart methods on a wide range of benchmark datasets. [ABSTRACT FROM AUTHOR]

• This paper proposes a K-CP method to learn codebooks for tensor training samples. • Sparse coefficients of samples are calculated as feature representations. • The method is applied to extract spatiotemporal features of multi-phase CT images. • The extracted features are used in classification of focal liver lesions. • Higher classification accuracy was achieved than conventional methods. Medical images play an important role in clinics. In most clinic sites, the diagnosis of diseases and the comprehending of disease progression need firstly accurate interpretation of the available medical images, which would be time-consuming in manual interpretation of the accumulated large amount of medical images. Thus automatical analysis and understanding of the available medical images become an active research topic, and therein, feature extraction of medical images plays an important role for achieving diagnosis performance. Natural images hold two-dimensional structures and linear statistical methods, such as k -means, GMM, and sparse coding, are widely applied for extracting compact and inherent representations. In contrast, medical images themselves have three-dimensional structures and possibly consist of multi-phase extension. Directly applying linear methods on the available medical data would lead to high dimensional vectors by reshaping the multi-dimensional data and would destroy the correlated relation among different dimensions of the raw medical data domain. Therefore, this study proposes a multilinear extension of the linear sparse coding for extracting compact and effective intermediate representations of the multi-dimensional local structures in multi-phase CT images, and aggregating the intermediate representations in the Bag-of-Visual-Words (BoVW) manner for classification of focal liver lesions (FLLs). In the proposed approach, three-layer volumes from the corresponding slices of multi-phase CT images are formed and spatiotemporal local structures from the volumes are extracted as 3rd-order tensors. Regarding the high dimensional local structures as tensors, we propose a K-CP (CANDECOMP/PARAFAC) algorithm to learn a tensor dictionary in an iterative way and extract the sparse representation with a multilinear OMP method. The aggregation of the sparse representation is implemented in the BoVW manner, which has been proved to be an effective method for extracting features from natural images and medical images. The proposed strategy is evaluated in classification of focal liver lesions and achieved better results than conventional BoVW with linear statistical methods. [ABSTRACT FROM AUTHOR]

Steganographic schemes are used in many fields, as it provides high-security transmission of messages. The strength of the steganographic scheme depends on embedding efficiency. Embedding efficiency is understood as the average number of random data bits that must be embedded in an object per one embedding change. The below-done work introduces an efficient embedding technique using various linear block codes useful to both spatial and frequency domain of digital images. Linear block coding algorithms like binary hamming code, Random linear code, Cyclic code, Reed Solomon code, etc., are applied to embed restricted data inside the image, which improves embedding efficiency. Higher embedding efficiency translates to better steganographic security. The Mean squared error (MSE), Peak Signal to noise ratio (PSNR), Structural Similarity Index (SSIM), Normalized Absolute Error (NAE) and Correlation Coefficient values for each algorithm is calculated and compared. Histogram attack is one of the steganalysis techniques which is used in this paper to confirm the security provided to the restricted data. This comparison shows that Binary Hamming code in the spatial domain and transform domain provides better results as compared to the other coding theory algorithms. [ABSTRACT FROM AUTHOR]

Minimal codes are a special type of linear codes, which have nice applications in secret sharing and secure two-party computation. How to construct new infinite family of minimal codes has been a hot topic in coding theory and cryptography. In this paper, by employing exponential sums, we study the Lee-weight distributions of several classes of linear codes from defining sets over the finite chain ring F p + u F p and determine exactly the complete weight enumerator of their Gray images under the Gray map. Furthermore, we prove that the Gray images of these linear codes are new infinite families of minimal five-weight linear codes with w min w max < p − 1 p. [ABSTRACT FROM AUTHOR]

This paper focuses on a linear code C with all symbol locality such that its code symbols set is partitioned to s disjoint components and every code symbol in each component j has t -pairwise disjoint recovery sets of size r j , for j ∈ { 1 , 2 , ... , s } , where r 1 , ... , r s and t are given integers. Code C is a locally recoverable code with multiple (r j) j ∈ [ s ] -localities and availability t and is defined by t − M L s − L R C. We present an upper bound on minimum Hamming distance d for s ≥ 2 and explain the result of the new bound using an example. [ABSTRACT FROM AUTHOR]

In this paper, we describe a correspondence between a fuzzy linear code and a family of nested linear codes. We also describe the arithmetic of fuzzy linear codes. As a special class of nested linear codes, we consider a family of nested self-orthogonal codes. A linear code is self-orthogonal if it is contained in its dual and self-dual if it is equal to its dual. We introduce a definition of fuzzy self-dual or self-orthogonal codes which include classical self-dual or self-orthogonal codes. As examples, we construct several interesting classes of fuzzy linear codes including fuzzy Hamming codes, fuzzy Golay codes, and fuzzy Reed-Muller codes. We also give a general decoding algorithm for fuzzy linear codes. [ABSTRACT FROM AUTHOR]

• Text-dependent speaker verification means the process of verifying the claimed identity of a person based on the pre-defined utterances for the system. • This can be done using the standard procedure or a novel method called one-for-all classification. • In this type, the difference between the features and reference values will be used and only one binary classification will be performed for all speakers of the dataset. • Significant improvements in system performance and an increase in classification duration are the results of using this method. Presenting a system for verifying the identity of people so that the data could be recorded in the simplest possible way, performing identity verification with acceptable accuracy, and being robust to attacks and noises is one of some important issues in today's era. This paper presents a text-dependent speaker verification system in which features are extracted from the data using a method called discrete wavelet transform based on linear prediction coding. In addition to the conventional classification in these systems, a new method for classification in which one binary classification will perform for all speakers of the dataset can also be done, which will be introduced in this research. Then the performance of the designed system on ASVspoof 2015, ASVspoof 2017, ASVspoof 2019, AudioMNIST, and TIMIT datasets will be evaluated using accuracy, precision, sensitivity, specificity, F1-score, and equal error rate (EER) metrics. It will be shown that the EER for the case where classification is done once for each speaker is 0.14 ± 0.83 % for the ASVspoof 2017 dataset and the accuracy and EER for the case where classification is done once for all speakers is 100 ± 0.00 % and 0.00 ± 0.00 % for all datasets. [ABSTRACT FROM AUTHOR]

Bose-Chaudhuri-Hocquenghem (BCH) codes have been intensively investigated. Even so, there is only a little known about primitive BCH codes, let alone non-primitive ones. In this paper, let q > 2 be a prime power, the dimension of a family of non-primitive BCH codes of length n = q m + 1 (also called antiprimitive) is studied. These codes are also linear codes with complementary duals (called LCD codes). Through some approaches such as iterative algorithm, partition and scaling, all coset leaders of C x modulo n with q ⌈ m 2 ⌉ < x ≤ 2 q ⌈ m 2 ⌉ + 2 are given for m ≥ 4. And for odd m the first several largest coset leaders modulo n are determined. Furthermore, a new kind of sequences is introduced to determine the second largest coset leader modulo n with m even and q odd. Also, for even m some conjectures about the first several coset leaders modulo n are proposed, whose complete verification would wipe out the difficult problem to determine the first several coset leaders of antiprimitive BCH codes. After deriving the cardinalities of the coset leaders, we shall calculate exact dimensions of many antiprimitive LCD BCH codes. [ABSTRACT FROM AUTHOR]

An [ n , k , d ] linear code is said to be maximum distance separable (MDS) or almost maximum distance separable (AMDS) if d = n − k + 1 or d = n − k , respectively. If a code and its dual code are both AMDS, then the linear code is called a near maximum distance separable (NMDS) code. NMDS codes correspond to some interesting objects in finite geometry and have nice applications in cryptography. The NMDS codes constructed from elliptic curves are referred as elliptic curve NMDS codes. In this paper, by adding two suitable columns to the generator matrices of famous Reed-Solomon codes, we can obtain two families of NMDS codes with parameters [ q + 3 , 4 , q − 1 ] q (q is a prime power with gcd ⁡ (q − 1 , 3) = 1) and [ 2 m + 3 , 5 , 2 m − 2 ] 2 m respectively. These NMDS codes are shown to be linearly inequivalent to elliptic curve NMDS codes. Besides, the weight enumerators and locality of these NMDS codes are completely determined. It turns out that the resultant NMDS codes and their dual codes are mostly distance-optimal and dimension-optimal locally recoverable codes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Toeplitz codes of index t , which are a generalization of quasi-cyclic codes of index t. We show that such codes meet the asymptotic Gilbert-Varshamov bound on the relative distance with asymptotic rate 1 t over F q , so they are asymptotically good for any t. We characterize some special classes of Toeplitz codes as LCD codes (resp. linear codes with one-dimensional hull or self-orthogonal codes). It is worth mentioning that we determine the hull dimension of the isometric image of a linear code. Some optimal or almost optimal binary linear codes are obtained as a result. Most importantly, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) from quasi-cyclic codes. In particular, we construct binary LCD [ 32 , 16 , 8 ] , [ 33 , 11 , 11 ] and [ 34 , 17 , 8 ] codes with improved parameters directly. [ABSTRACT FROM AUTHOR]

Twisted generalized Reed-Solomon codes are the generalization of generalized Reed-Solomon codes, due to their good algebraic structure and applications in cryptography, which have been widely studied in recent years. As a general algebraic decoding method for linear codes, the error-correcting pair exists in many classical linear codes. In this paper, we extend the error-correcting pair to twisted generalized Reed-Solomon codes. For twisted generalized Reed-Solomon codes with twist 1 or 2, we constructively show that the existence of the error-correcting pair. Moreover, we introduce the corresponding decoding procedure of twisted generalized Reed-Solomon codes by ℓ -error-correcting pairs and give several examples. [ABSTRACT FROM AUTHOR]

Abstract: Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have very efficient encoding and decoding algorithms. In this paper, monomials and trinomials over finite fields with even characteristic are employed to construct a number of families of binary cyclic codes. Lower bounds on the minimum weight of some families of the cyclic codes are developed. The minimum weights of other families of the codes constructed in this paper are determined. The dimensions of the codes are flexible. Some of the codes presented in this paper are optimal or almost optimal in the sense that they meet some bounds on linear codes. Open problems regarding binary cyclic codes from monomials and trinomials are also presented. [Copyright &y& Elsevier]

For any admissible value of the parameters n and k there exist [ n , k ] -Maximum Rank Distance F q -linear codes. Indeed, it can be shown that if field extensions large enough are considered, almost all rank metric codes are MRD. On the other hand, very few families up to equivalence of such codes are currently known. In the present paper we study some invariants of MRD codes and evaluate their value for the known families, providing a new characterization of generalized twisted Gabidulin codes. [ABSTRACT FROM AUTHOR]

Locality-constrained Linear Coding (LLC) based visual tracking can give a better and faster tracking performance than traditional sparse representation based tracking methods. However, the existing LLC based methods often use the anchor points near the target to build the sparse coding dictionary for local sparse coding. It may cause a problem that it is hard to discriminate the difference between the negative and positive anchor points in sparse coding dictionary when facing severe background clutter, illumination change and occlusion. In this paper, we propose a context aware sparse coding method to achieve robust visual tracking. The proposed method can prevent the negative anchor points from disturbing the classifier accuracy because it uses a global context regularizer to constrain the sparse coding value of those negative anchor points that are similar to the positive anchor points. Experiment results show that our method can achieve a better tracking performance than state-of-the-art tracking methods do. [ABSTRACT FROM AUTHOR]

Abstract Ding (2016) introduced a generic construction of linear codes over F p from a subset of F p m . Many combinatorially interesting sets give rise to codes with good parameters under this generic construction. Ding also provided a set of interesting conjectures related to the properties of several sets arising from vectorial Boolean functions and their corresponding linear codes under this construction. In this paper we prove four of these conjectures (Conjectures 38–41). [ABSTRACT FROM AUTHOR]

Highlights • Convolution-deconvolution auto encoder is introduced for process monitoring. • No assumptions or requirements are needed for input data of batch processes. • Autocorrelations and dynamic cross-correlations of the variables are considered. • Improved model generalization abilities under various process conditions. • Superior solutions for fault detection and process condition identification. Abstract Feature learning is a generic and fundamental problem in data-based process monitoring of batch processes, such as injection molding. This paper proposes an automatic feature learning method, considering the following two vital characteristic aspects: the mechanical characteristics within a batch and the inherent characteristics among batches. Unsupervised feature learning is performed using convolution-deconvolution auto encoders, and the learned features are applied as predefined parameters for process monitoring, including process condition identification and fault detection. Experiments are carried out with different process conditions. The results indicate that the proposed method achieves improved model generalization ability under various process conditions, which means it can precisely reveal the variable autocorrelations and cross-correlations among different variables. Meanwhile, the learned features achieve higher classification accuracies and offer more optimal solutions for process monitoring. This method has been proven as an efficient means of feature learning, which should be appropriate for other batch processes. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

A Reynolds-averaged Navier-Stokes solver has been equipped with two different transition prediction and modeling techniques for automatic transition prediction on general three-dimensional geometrical configurations. The first technique is a streamline-based approach that applies an automated local linear stability code, the eN-method and a simplified two-N-factor strategy and is applicable with any kind of turbulence model. The second technique is a transition transport equation approach and uses the γ -Re θ ,t model. The standard γ -Re θ ,t model has been extended by a modeling approach that captures transition due to crossflow instabilities. The extended model can be applied to three-dimensional configuration exhibiting crossflow transition whereas the original γ -Re θ ,t model only captures streamwise transition mechanisms. The paper focuses on a three-dimensional inclined 6:1 prolate spheroid which can be considered as a simplified fuselage configuration. For a number of years this configuration has become a standard basic three-dimensional test case for transition prediction approaches and has been used in a number of international workshops and research activities. The focus of the paper is the validation of the two transition modeling approaches for six different angles of attack using two different turbulence models, a standard two-equation eddy-viscosity model and a differential Reynolds-stress model. The simulation results from the two approaches have been compared against each other and to the available experimental data. The comparisons highlight as many qualitative and quantitative similarities and agreements as they reveal quantitative differences and deviations of specific details. [ABSTRACT FROM AUTHOR]