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The classical way of extending an [ n , k , d ] linear code C is to add an overall parity-check coordinate to each codeword of the linear code C. This extended code, denoted by C ‾ (− 1) and called the standardly extended code of C , is a linear code with parameters [ n + 1 , k , d ¯ ] , where d ¯ = d or d ¯ = d + 1. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper. [ABSTRACT FROM AUTHOR]

The central levels problem asserts that the subgraph of the (2 m + 1) -dimensional hypercube induced by all bitstrings with at least m + 1 − ℓ many 1s and at most m + ℓ many 1s, i.e., the vertices in the middle 2 ℓ levels, has a Hamilton cycle for any m ≥ 1 and 1 ≤ ℓ ≤ m + 1. This problem was raised independently by Buck and Wiedemann, Savage, Gregor and Škrekovski, and by Shen and Williams, and it is a common generalization of the well-known middle levels problem , namely the case ℓ = 1 , and classical binary Gray codes, namely the case ℓ = m + 1. In this paper we present a general constructive solution of the central levels problem. Our results also imply the existence of optimal cycles through any sequence of ℓ consecutive levels in the n -dimensional hypercube for any n ≥ 1 and 1 ≤ ℓ ≤ n + 1. Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the n -dimensional hypercube, n ≥ 2 , that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code. [ABSTRACT FROM AUTHOR]

Linear complementary dual (LCD) codes provide an optimum linear coding solution for the two-user binary adder channel. LCD codes also can be used against side-channel attacks and fault non-invasive attacks. In this paper, we obtain a lower bound on the distance of binary LCD codes through expanded codes. We give necessary and sufficient conditions to extend binary [ n , k ] LCD codes to binary [ n + 1 , k ] and [ n + 1 , k + 1 ] LCD codes. A sufficient condition for a binary [ n , k , d − 1 ] LCD code constructed from a binary [ n + 1 , k , d ] LC D o , e code is also given. Finally, we construct some new binary LCD codes by using our LCD bounds and some methods such as puncturing, shortening, expanding and extension. In particular, we improve some previously known range of d LCD (n , k) of lengths 38 ≤ n ≤ 40 and dimensions 9 ≤ k ≤ 15. We also obtain some values or range of d LCD (n , k) with 41 ≤ n ≤ 50 and 6 ≤ k ≤ n − 6. [ABSTRACT FROM AUTHOR]

This paper presents a topology constrained phase field model based on a variable mobility Cahn-Hilliard equation. Mostly in the level set framework, this issue has been addressed using concepts from digital topology. However, point based specifications can not be effective in phase fields where the interface is represented by a diffuse region. In phase field models, topology constraints are included in the energy as a penalty term. However, there are no studies which provide topological constraints in the Cahn-Hilliard model to our knowledge. In this work, we use the mobility parameter to enforce topology constraints; this allows local, explicit control of topological events, unlike the optimization based solutions. We define topologically critical regions in phase fields based on a topological analysis of phase field evolution. This analysis shows that diffuse layers formed at topological encounters behave like a Morse function and the critical points of these transitional layers give topologically critical points in phase fields. Using this property, we first detect critical points using an efficient image analysis method. Then we detect the transitional region (critical region) around each critical point using a local morphological segmentation algorithm. We define a mobility function which both has a spatial and concentration dependence. This function is designed to reverse the effect of diffuse layers formed at topological encounters and prevent the further development of topological events. Being extracted from phase field values, the mobility map naturally aligns with phase field evolution. Together with the use of an unconditionally stable semi-implicit Fourier spectral method for the variable mobility Cahn-Hilliard equation, an efficient model is provided. Local, explicit control mechanism allows the design of free energy functional for modeling different materials; this extends the use of Cahn-Hilliard equation to applications which require topology control. • Topologically critical regions in phase fields. • Critical region detection using a local segmentation algorithm (seed based region growing). • Topological analysis of phase fields (critical point detection) using Local Binary Pattern codes. • Concentration dependent variable mobility for topology control in phase field equations. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in Tang and Ding (2022) [26] and the works in Liu et al. (2023) [15] and Liu et al. (2023) [16] , the objectives of this paper are the construction and analysis of four infinite families of ternary cyclic codes with length n = 3 m − 1 for odd m and dimension k ∈ { n / 2 , (n + 2) / 2 } whose minimum distances have a square-root-like lower bound. Their duals have parameters [ n , k ⊥ , d ⊥ ] , where k ⊥ ∈ { n / 2 , (n − 2) / 2 } and d ⊥ also has a square-root-like lower bound. These families of codes and their duals contain distance-optimal cyclic codes. [ABSTRACT FROM AUTHOR]

This paper proposes a methodology for developing multi-site damage location systems for engineering structures that can be trained using single-site damaged state data only. The methodology involves training a sequence of binary classifiers based upon single-site damage data and combining the developed classifiers into a robust multi-class damage locator. In this way, the multi-site damage identification problem may be decomposed into a sequence of binary decisions. In this paper Support Vector Classifiers are adopted as the means of making these binary decisions. The proposed methodology represents an advancement on the state of the art in the field of multi-site damage identification which require either: (1) full damaged state data from single- and multi-site damage cases or (2) the development of a physics-based model to make multi-site model predictions. The potential benefit of the proposed methodology is that a significantly reduced number of recorded damage states may be required in order to train a multi-site damage locator without recourse to physics-based model predictions. In this paper it is first demonstrated that Support Vector Classification represents an appropriate approach to the multi-site damage location problem, with methods for combining binary classifiers discussed. Next, the proposed methodology is demonstrated and evaluated through application to a real engineering structure – a Piper Tomahawk trainer aircraft wing – with its performance compared to classifiers trained using the full damaged-state dataset. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in Tang and Ding (2022) [14] , the objectives of this paper are the construction and analyses of five infinite families of binary cyclic codes with parameters [ n , k ] and (n − 6) / 3 ≤ k ≤ 2 (n + 6) / 3. Three of the five families of binary cyclic codes and their duals have a very good lower bound on their minimum distances and contain distance-optimal codes. The other two families of binary cyclic codes are composed of binary duadic codes with a square-root-like lower bound on their minimum distances. As a by-product, two infinite families of self-dual binary codes with a square-root-like lower bound on their minimum distances are obtained. [ABSTRACT FROM AUTHOR]

A pair (n , q) with q a prime power and n a positive integer for which gcd ⁡ (n , q) = 1 is non-standard if one of the following properties holds, which we will show to be equivalent. − The group U n , q of n -th roots-of-unity in the splitting field F q m of x n − 1 over F q is fixed set-wise by an F q -linear map on F q m that is not of the form x ⟶ α x q j . − There is a non-cyclic linear recurring sequence s of period n (so with s not of the form s i = α ξ i for i ≥ 0) with associated characteristic polynomial being irreducible over F q , such that U n , q = { s 0 , ... , s n − 1 }. In that case, the group U n , q is called non-standard by Brison and Nogueira, who studied this phenomenon in a sequence of papers. − A q -ary irreducible cyclic code of length n has "extra" permutation automorphisms (some well-known examples are the (duals of the) Golay codes and binary simplex codes for n ≥ 7). We will refer to such codes as non-standard irreducible cyclic codes or NSIC-codes. We first investigate non-standard linear recurring sequence subgroups and establish the equivalence between the first and second of the above properties. Then we investigate permutation automorphisms of general (repeated-root) cyclic codes and irreducible cyclic codes and establish the equivalence between the first and third of these properties. We also introduce a notion of non-standardness for general cyclic codes, and relate it to our earlier definition of NSIC-codes. This paper is the first part of an expanded version of the arXiv paper [28]. [ABSTRACT FROM AUTHOR]

In this paper, we give an improved bound for mixed exponential sums associated to good cyclic codes. [ABSTRACT FROM AUTHOR]

Abstract We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F 2 + u F 2 , using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper. Highlights • A novel bordered construction for self-dual codes can be obtained using groups rings. • Extremal binary self-dual codes of different lengths can be constructed using the constructions. • 41 new additions to the set of known extremal binary self-dual codes of length 68 have been made. • There is a strong connection between groups rings and constructions for self-dual codes. [ABSTRACT FROM AUTHOR]

Abstract Perceptual quality evaluation of multiply distorted images has become a very challenging research topic. In this paper, we present a novel and efficient deep blind quality evaluator for multiply distorted images based on monogenic binary coding (MBC). Local complementary structural information and a deep learning method are employed to blindly evaluate the quality of multiply distorted images. First, a monogenic signal representation is utilized to decompose a multiply distorted image into three complementary components: orientation, phase, and magnitude. The quality-predictive features are then determined from the complementary components. Finally, the features are mapped to the human quality score of the multiply distorted image based on the deep neural network. The results on two newly established multiply distorted image subjective databases confirm that our metric has a better prediction performance than existing state-of-the-art full-reference and classical blind metrics. [ABSTRACT FROM AUTHOR]

Abstract This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If RG is a group ring, where R is commutative and S is a set of generators of G then necessary and sufficient conditions on a map from S to RG are established, such that the map can be extended to an R -derivation of RG. Derivations are shown to be trivial for semisimple group algebras of abelian groups. The derivations of finite group algebras are constructed and listed in the commutative case and in the case of dihedral groups. In the dihedral case, the inner derivations are also classified. Lastly, these results are applied to construct well known binary codes as images of derivations of group algebras. [ABSTRACT FROM AUTHOR]

Abstract Under research for nearly sixty years, Bose–Chaudhuri–Hocquenghem (BCH) codes have played increasingly important roles in many applications such as communication, data storage and information security. However, the dimension and minimum distance of BCH codes have been seldom solved by now because of their intractable characteristics. The objective of this paper is to study the dimensions of some binary BCH codes with length n = 2 m + 1. Many new techniques are employed to investigate the coset leaders modulo n. For m = 2 t + 1 , 4 t + 2 , 8 t + 4 and m ≥ 10 , the first five largest coset leaders modulo n are determined, and the dimensions of some BCH codes of length n with designed distance δ > 2 ⌈ m 2 ⌉ are presented. These new skills and results may be helpful to study other types of cyclic codes over finite fields. [ABSTRACT FROM AUTHOR]

Abstract In this paper we give new constraints on the distance distribution of doubly constant-weight (binary) codes. These constraints improve the linear programming bound on sizes of doubly constant-weight codes. Computations are done for all codes of length n ≤ 28 and all improved upper bounds are shown. We moreover show that the improved upper bounds give rise to further new upper bounds. [ABSTRACT FROM AUTHOR]

Maximum-distance-separable (MDS) codes and self-dual codes are important families of classical codes in coding theory. It is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field F q has been completely solved for q is even. In this paper, for finite field with odd characteristic, we obtain some new results on the existence of MDS self-dual codes by generalized Reed-Solomon (GRS) codes and extended GRS codes. [ABSTRACT FROM AUTHOR]

Binary odd-like duadic codes have parameters [ n , (n + 1) / 2 ] , where n is odd. It is well known that the minimum odd weight of odd-like duadic codes has the lower bound n. The binary quadratic-residue codes and the punctured binary Reed-Muller codes of certain order are two families of binary cyclic codes with parameters [ n , (n + 1) / 2 , d ≥ n ]. It is very hard to construct an infinite family of binary cyclic codes with length n and dimension near (n + 1) / 2 whose minimum distances have a square-root bound. Recently, Tang and Ding constructed an infinite family of binary cyclic codes with parameters [ 2 m − 1 , 2 m − 1 , d ] whose minimum distances have lower bounds close to the square-root bound. The objective of this paper is to construct several infinite families of binary cyclic codes of length 2 m − 1 and dimension near 2 m − 1 whose minimum distance d much exceeds the square-root bound. For m ≥ 9 being odd, an infinite family of binary cyclic codes with parameters [ 2 m , 2 m − 1 , d ≥ 3 ⋅ 2 m − 1 2 − 1 ] is presented. For m ≥ 8 being even, an infinite family of binary cyclic codes with parameters [ 2 m , 2 m − 1 − 1 + m 2 , d ≥ 3 ⋅ 2 m − 2 2 + 1 ] is presented. [ABSTRACT FROM AUTHOR]

Linear codes and t -designs are interactive with each other. It is well known that some t -designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended codes of linear codes holding an infinite family of t -designs with t ≥ 3 are reported in the literature. In this paper, we study the extended codes of the augmented codes of a class of binary cyclic codes with three zeros and their dual codes, and show that those codes hold 3-designs. Furthermore, we obtain some shortened codes from the studied cyclic codes and explicitly determine their parameters. Some of those shortened codes are optimal or almost optimal. [ABSTRACT FROM AUTHOR]

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over F q m are GRS codes (resp., extended GRS codes) when m = 1. In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs. [ABSTRACT FROM AUTHOR]

Let n ∈ N , R be a binary relation on [ n ] , and C 1 ( i , j ) , … , C n ( i , j ) ∈ Z , for i , j ∈ [ n ] . We define the exponential system of equations E ( R , ( C k ( i , j ) i , j , k ) to be the system X i Y 1 C 1 ( i , j ) ⋯ Y n C n ( i , j ) = X j , for ( i , j ) ∈ R , in variables X 1 , … , X n , Y 1 , … , Y n . The aim of this paper is to classify precisely which of these systems admit a monochromatic solution ( X i , Y i ≠ 1 ) in an arbitrary finite colouring of the natural numbers. This result could be viewed as an analogue of Rado's theorem for exponential patterns. [ABSTRACT FROM AUTHOR]

This paper studies and classifies optimal binary self-dual codes having an automorphism of order 7 with 9 cycles. This classification is done by applying a method for constructing binary self-dual codes with an automorphism of odd prime order p . There are exactly 69781 inequivalent binary self-dual [ 64 , 32 , 12 ] codes with an automorphism of type 7 − ( 9 , 1 ) . As for binary [ 66 , 33 , 12 ] self-dual codes with an automorphism of type 7 − ( 9 , 3 ) there are 1652432 such codes. We also construct more than 4 million new optimal codes of length 68 among which are the first known examples of the very elusive s -extremal self-dual codes. We prove the nonexistence of [ 70 , 35 , 14 ] codes with an automorphism of type 7 − ( 9 , 7 ) . Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before. [ABSTRACT FROM AUTHOR]

The locality of locally repairable codes (LRCs) for a distributed storage system is the number of nodes that participate in the repair of failed nodes, which characterizes the repair cost. In this paper, we first determine the locality of MacDonald codes, then propose three constructions of LRCs with r = 1 , 2 and 3 . Based on these results, for 2 ≤ k ≤ 7 and n ≥ k + 2 , we give an optimal linear [ n , k , d ] code with small locality. The distance optimality of these linear codes can be judged by the codetable of M. Grassl for n < 2 ( 2 k − 1 ) and by the Griesmer bound for n ≥ 2 ( 2 k − 1 ) . Almost all the [ n , k , d ] codes ( 2 ≤ k ≤ 7 ) have locality r ≤ 3 except for the three codes, and most of the [ n , k , d ] code with n < 2 ( 2 k − 1 ) achieves the Cadambe–Mazumdar bound for LRCs. [ABSTRACT FROM AUTHOR]

Binary constant weight codes have important applications in various topics and have been studied for many years. Optimal or near-optimal binary constant weight codes of small lengths have been determined. In this paper we propose an improvement of the Ericson–Zinoviev construction of binary constant weight codes from q -ary codes. By applying this improvement to Reed–Solomon codes, some new or optimal binary constant weight codes are presented. In particular new binary constant weight codes A ( 64 , 10 , 8 ) ≥ 4112 and A ( 64 , 12 , 8 ) ≥ 522 are constructed. We also give explicitly constructed binary constant weight codes which improve the Gilbert and the Graham–Sloane lower bounds asymptotically in a small range of parameters. Some new binary constant weight codes constructed from algebraic-geometric codes by applying our this improvement are also presented. [ABSTRACT FROM AUTHOR]

The paper begins by giving a counter example to show that the algorithm for construction of new perfect poset codes from a given perfect poset code by removal of a coordinate as given by Lee (2004) [11] does not hold. The algorithm has been improved and generalized to obtain new perfect poset block codes from a given perfect poset block code. The modified necessary and sufficient conditions for the construction of new perfect poset codes have been derived as a particular case. A bound has been obtained on the height of poset P s that turns a given π -code into r -perfect ( P s , π ) -code. We show that there does not exist a poset which admits the binary Simplex code of order 3 to be a 2-perfect poset code. Also, all the poset structures which admit the extended ternary Golay code to be a 3-perfect poset code have been classified. [ABSTRACT FROM AUTHOR]

The critical threshold of a (simple binary) matroid N is the infimum over all ρ such that any N -free matroid M with | M | > ρ 2 r ( M ) has bounded critical number. In this paper, we resolve two conjectures of Geelen and Nelson, showing that the critical threshold of the projective geometry PG ( t − 1 , 2 ) is 1 − 3 ⋅ 2 − t . We do so by proving the following stronger statement: if M is PG ( t − 1 , 2 ) -free with | M | > ( 1 − 3 ⋅ 2 − t ) 2 r ( M ) , then the critical number of M is t − 1 or t . Together with earlier results of Geelen and Nelson [9] and Govaerts and Storme [11] , this completes the classification of dense PG ( t − 1 , 2 ) -free matroids. [ABSTRACT FROM AUTHOR]

Let r i be positive integers and R i = Z 2 [ x ] / 〈 x r i − 1 〉 for 1 ≤ i ≤ ℓ . Denote R = R 1 × R 2 × ⋯ × R ℓ . Generalized quasi-cyclic (GQC) code C of length ( r 1 , r 2 , … , r ℓ ) over Z 2 can be viewed as Z 2 [ x ] -submodule of R . In this paper, we investigate the algebraic structure of C by presenting its normalized generating set. We also present a method to determine the normalized generating set of the dual code of C , which is derived from the normalized generating set of C . [ABSTRACT FROM AUTHOR]

A code C in the Hamming metric, that is, is a subset of the vertex set V Γ of the Hamming graph Γ = H (m , q) , gives rise to a natural distance partition { C , C 1 , ... , C ρ } , where ρ is the covering radius of C. Such a code C is called completely transitive if the automorphism group Aut (C) acts transitively on each of the sets C , C 1 , ..., C ρ. A code C is called 2-neighbour-transitive if ρ ⩾ 2 and Aut (C) acts transitively on each of C , C 1 and C 2. Let C be a completely transitive code in a binary (q = 2) Hamming graph having full automorphism group Aut (C) and minimum distance δ ⩾ 5. Then it is known that Aut (C) induces a 2-homogeneous action on the coordinates of the vertices of the Hamming graph. The main result of this paper classifies those C for which this induced 2-homogeneous action is not an affine, linear or symplectic group. We find that there are 13 such codes, 4 of which are non-linear codes. Though most of the codes are well-known, we obtain several new results. First, a non-linear completely transitive code that does not explicitly appear in the existing literature is constructed, as well as a related non-linear code that is 2-neighbour-transitive but not completely transitive. Moreover, new proofs of the complete transitivity of several codes are given. Additionally, we consider the question of the existence of distance-regular graphs related to the completely transitive codes appearing in our main result. [ABSTRACT FROM AUTHOR]

Recently, linear codes with few weights have been studied extensively. These linear codes have wide applications in secret sharing schemes and authentication codes. In this paper, we introduce a new construction of defining sets using cyclotomic mappings and construct three new classes of binary linear codes with two or three weights. We also explicitly determine the weight distributions of these codes. [ABSTRACT FROM AUTHOR]

An additive cyclic code of length n over F 4 can be defined equivalently as an F 2 [ x ] / 〈 x n + 1 〉 -submodule of F 4 [ x ] / 〈 x n + 1 〉. In this paper we study additive cyclic and complementary dual codes of odd length over F 4 with respect to the trace Hermitian inner product and the trace Euclidean inner product. We characterize subfield subcodes and trace codes of these codes by their generators as binary cyclic codes. [ABSTRACT FROM AUTHOR]

In this paper, we prove that an even rootless lattice L is isomorphic to 2 A 1 , 2 D 4 , 2 E 8 , or BW 16 if the lattice type vertex operator algebra V L + is of class S 4 . In addition, we prove that the vertex operator algebra V 2 D 4 + is of class S 5 , and the vertex operator algebras V 2 A 1 + , V 2 E 8 + , and V BW 16 + are of class S 7 . [ABSTRACT FROM AUTHOR]

Let R = Z 4 be the integer ring mod 4. A double cyclic code of length ( r , s ) over R is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as R [ x ] -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . In this paper, we determine the generator polynomials of this family of codes as R [ x ] -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . Further, we also give the minimal generating sets of this family of codes as R -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual. [ABSTRACT FROM AUTHOR]

Two-weight linear codes have many wide applications in authentication codes, association schemes, strongly regular graphs, and secret sharing schemes. In this paper, we present two classes of two-weight binary or ternary linear codes. In some cases, they are optimal or almost optimal. They can also be used to construct secret sharing schemes. [ABSTRACT FROM AUTHOR]

This paper presents a novel, simple, yet robust texture descriptor against noise named the adjacent evaluation local binary patterns (AELBP) for texture classification. In the proposed approach, an adjacent evaluation window is constructed to modify the threshold scheme of LBP. The neighbors of the neighborhood center g c are set as the evaluation center a p . Surrounding the evaluation center, we set up an evaluation window and calculate the value of a p , and then extract the local binary codes by comparing the value of a p with the value of the neighborhood center g c . Moreover, this adjacent evaluation method is generalized and can be integrated with the existing LBP variants such as completed local binary pattern (CLBP) and local ternary pattern (LTP) to derive new image features against noise for texture classification. The proposed approaches are compared with the state-of-the-art approaches on Outex and CUReT databases, and evaluated on three challenging databases (i.e. UIUC, UMD and ALOT databases) for texture classification. Experimental results demonstrate that the proposed approaches present a solid power of texture classification under illumination and rotation variations, significant viewpoint changes, and significant large-scale challenging conditions. Furthermore, the proposed approaches are more robust against noise and consistently outperform all the basic approaches in comparison. [ABSTRACT FROM AUTHOR]

In this paper, we give a method to lift binary self-dual codes to the ring R k . The lifting method requires solving a system of linear equations over R k . This technique is applied to [ 14 , 7 , 4 ] binary self-dual code to obtain self-dual codes over R 2 . As Gray images of these codes, a substantial number of [ 56 , 28 , 10 ] self-dual codes are generated. By using the extension theorem given by Bouyuklieva and Bouyukliev in [2] , ten new extremal binary self-dual codes of length 58 with new enumerators are found which were not previously known to exist. [ABSTRACT FROM AUTHOR]

For two positive integers m and k , let C e be a class of cyclic code of length 2 m − 1 over F 2 with three nonzeros γ − 1 , γ − ( 2 k + 1 ) and γ − ( 2 e k + 1 ) for e = 2 or 3, where γ is a primitive element of F 2 m . When m gcd ⁡ ( m , k ) is odd, Kasami in 1971 determined the weight distributions of cyclic codes C 2 and C 3 , which is the same as that of the dual of the triple error-correcting BCH code. This paper obtains the weight distributions of C 2 and C 3 for the case of m gcd ⁡ ( m , k ) being even. [ABSTRACT FROM AUTHOR]

Recently, techniques that can automatically figure out the incisive information from gigantic visual databases are urging popularity. The existing multi-feature hashing method has achieved good results by fusing multiple features, but in processing these multi-features, fusing multi-features into one feature will cause the feature dimension to be very high, increasing the amount of calculation. On the one hand, it is not easy to discover the internal ties between different features. This paper proposes a novel unsupervised multiple feature hashing for image retrieval and indexing (MFHIRI) method to learn multiple views in a composite manner. The proposed scheme learns the binary codes of various information sources in a composite manner, and our scheme relies on weighted multiple information sources and improved KNN concept. In particular, here we adopt an adaptive weighing scheme to preserve the similarity and consistency among binary codes. Precisely, we follow the graph modeling theory to construct improved KNN concept, which further helps preserve different statistical properties of individual sources. The important aspect of improved KNN scheme is that we can find the neighbors of a data point by searching its neighbors' neighbors. During optimization, the sub-problems are solved in parallel which efficiently lowers down the computation cost. The proposed approach shows consistent performance over state-of-the-art (three single-view and eight multi-view approaches) on three broadly followed datasets viz. CIFAR-10, NUS-WIDE and Caltech-256. • Weight learning scheme is adopted to combine the views to confine the deviation among views. • With the help of improved KNN scheme it is easy to find the neighbors' neighbors of a datapoint. • Proposed MFHIRI scheme is quite effective in training and viable in query time. • Optimize the correlation b/w learning continuous representations and consistent binarization. [ABSTRACT FROM AUTHOR]

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers. [ABSTRACT FROM AUTHOR]

In this paper, for some fixed q , we construct some new classes of maximum distance separable (MDS) self-dual codes over F q by using (extended) generalized Reed-Solomon codes. Our constructions utilize the additive and/or multiplicative structures of the finite field F q. It reveals that by choosing suitable subsets of F q , more MDS self-dual codes with new parameters can be found. [ABSTRACT FROM AUTHOR]

Two finite words u and v are called Abelian equivalent if each letter occurs equally many times in both u and v. The abelian closure A (x) of (the shift orbit closure of) an infinite word x is the set of infinite words y such that, for each factor u of y , there exists a factor v of x which is abelian equivalent to u. The notion of an abelian closure gives a characterization of Sturmian words: among binary uniformly recurrent words, Sturmian words are exactly those words for which A (x) equals the shift orbit closure Ω (x). In this paper we show that, contrary to larger alphabets, the abelian closure of a uniformly recurrent aperiodic binary word which is not Sturmian contains infinitely many minimal subshifts. [ABSTRACT FROM AUTHOR]

Linear codes with few weights have applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the butterfly structure [6,29] and the works of Li, Yue and Fu [21] and Jian, Lin and Feng [19] , we introduce a new defining set with the form of the closed butterfly structure and consequently we obtain three classes of 3-weight binary linear codes and a class of 4-weight binary linear codes whose dual is optimal. The lengths and weight distributions of these four classes of linear codes are completely determined by some detailed calculations on certain exponential sums. Computer experiments show that many (almost) optimal codes can be obtained from our construction. [ABSTRACT FROM AUTHOR]

Since online hashing has the advantages of low storage and fast calculation ,it attracts the attention of many scholars. However, the learning of new data streams separates the similarity between new data and existing data in many online hashing methods, which leads to poor retrieval performance. In addition, the similarity measure ignores the expression of different similarity. In this paper, we propose a novel supervised method, namely Label Projection Online Hashing for Balanced Similarity (LPOH). Compared with existing online hashing methods, LPOH aims to solve the problem of the effective establishment of the projection between the label vector and the binary code, and the successful realization of description of different similarity between the same labeled data. Specifically, LPOH overcomes the problem of similarity deviation caused by data imbalance via establishing a mapping matrix to derive a relationship between the data label vector and the binary code. Furthermore, the error between the binary code and the hash function concerning data streams is described. Extensive experiments on widely-used three benchmark datasets demonstrate that LPOH outperforms the state-of-the-art online hashing methods. [ABSTRACT FROM AUTHOR]

• A temporally-consistent local compact binary descriptor is used to capture background statistics. • The useful correlation exists in adjacent frames is exploited to guarantee high levels of foreground detection accuracy. • Multi-features histograms benefit from the merit of each feature. • The sensitivity and update rate of the model can be dynamically adjusted for diverse situations. This paper presents a moving object detection scheme that incorporates three innovations. First, considering the inter-frame consistency of pixels, we extend the compact binary face descriptor (CBFD) to the temporal domain and propose a novel local binary descriptor named temporally-consistent local compact binary descriptor (TC-LCBD), which exploits the useful correlation of the intensities of inter-frame pixels to guarantee good performance in complex scenes. We do this mainly because the background scene between frames has a significant coherence. Second, both color and TC-LCBD features are modeled as a group of adaptive histograms for characterizing each pixel, which can enhance the robustness to dynamic backgrounds and illumination changes. Third, by comparing changes in histogram proximity between two adjacent frames, we can dynamically adjust the model sensitivity and adaptation rate without user intervention. Experimental results on well-known, challenging data sets demonstrate that the proposed method significantly outperforms many state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of a linear code has many applications such as on the type II wire-tap channel, dealing with t -resilient functions and trellis or branch complexity of linear codes and so on. In this paper, we present a formula for computing the weight hierarchies of linear codes constructed by the generalized method of defining sets. Then, we construct two classes of binary linear codes with a few weights and determine their weight distributions and weight hierarchies completely. Some codes of them can be used in secret sharing schemes. [ABSTRACT FROM AUTHOR]

We derive bounds for the ball L p -discrepancies in the Hamming space for 0 < p < ∞ and p = ∞. Sharp estimates of discrepancies have been obtained for many spaces such as the Euclidean spheres and more general compact Riemannian manifolds. In the present paper, we show that the behavior of discrepancies in the Hamming space differs fundamentally because the volume of the ball in this space depends on its radius exponentially while such a dependence for the Riemannian manifolds is polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we propose a mechanism on how to construct long MDS self-dual codes from short ones. These codes are special types of generalized Reed-Solomon (GRS) codes or extended generalized Reed-Solomon codes. The main tool is utilizing additive structure or multiplicative structure on finite fields. By applying this method, more MDS self-dual codes can be constructed. [ABSTRACT FROM AUTHOR]