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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]

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]

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]

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]

Deep Hashing has achieved great success in large-scale image retrieval due to binary code's storage and computation efficiency. However, its learning paradigm under real-world environments is less studied, and most existing approaches are developed in the closed scenario, e.g., simple and unchanging semantics. When images of new classes emerge, they have to retrain the model on all history training datasets, but the constant data uploading makes this impractical. This paper proposes a novel method, called continual deep semantic hashing (CDSH), for learning binary codes of multi-label images with increasing classes. The CDSH consists of two hashing networks. One learns to hash the increasing semantics of data, i.e., label, into the semantic codes and accumulates label-code pairs as long-term knowledge, incorporating empirically verified loss and designed special regularization to ensure encoding old labels unchanged. The other learns to map images to the corresponding semantic code from a probabilistic view and solid knowledge via retaining history exemplar samples and projecting model gradients. We theoretically prove this improves the probability of old data's code unchanged after the model is updated. Extensive experiments on four widely used image datasets demonstrate that the CDSH method can continually learn hash functions and yield state-of-the-art retrieval performance. • The code drift is a key problem in continual hash learning for image retrieval. • The two-step hash learning is a useful architecture for continual hash learning. • Keeping label-code pairs can capture the similarities between old and new data. • The combination of different losses can alleviate the code drift of the hashing. • The gradient projection improves the probability of code of old data unchanged. [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]

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]

Camouflaged Object Detection (COD) aims to detect objects with high similarity to the background. Unlike general object detection, COD is a more challenging task because the target boundaries are vague and the location is difficult to determine. In this paper, we propose a novel COD framework, which consists of two main components, namely, Neighbor Connection Mode (NCM) and Hierarchical Information Transfer (HIT). NCM aggregates the features from the neighboring layers of the encoder network to enhance the complementation of various level information. Our NCM not only reduces the burden of dense connection that consumes a lot of computing memory and redundant features but also weakens the phenomenon of the long-term transmission of context. We also propose a HIT module to transfer the features of different dilated rates inside each level hierarchically, which expands the receptive field of each branch and enhances the relationship between different features. Our method accurately detects camouflaged objects by considering full level information and a large receptive field. The experiments on three COD datasets show that our model achieves state-of-the-art performance. • A double branch framework focuses on camouflaged object detection (COD). • A Neighbor Connection Mode (NCM) is proposed to aggregate adjacent level features. • Hierarchical Information Transfer (HIT) is presented to enlarge receptive fields. [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]