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In this paper, we construct new types of codes, called Weighted Block-wise Burst Error Correcting Codes (Weighted BBEC Codes), which are an improvement on the previously known block wise burst error correcting codes in the sense of their weight, i.e., (n = n1 + n2, k) linear codes that correct all the bursts of length b1 (fixed) with the weight w1(w1 ≤ b1) or more (less) in the first sub-block of length n1 and all the bursts of length b2 (fixed) with the weight w2 or more (less) in the next sub-block of length n2. [ABSTRACT FROM AUTHOR]

Linear complementary dual (LCD) codes, which are a class of linear codes introduced by Massey, have been extensively studied in the literature recently. It has been shown that LCD codes play a role in measures to counter passive and active side-channel analyses on embedded cryptosystems. In this paper, tables are presented of good quaternary Hermitian LCD codes and they are used in the construction of puncturing, shortening and combination codes. The results of this, including three tables of the best-known quaternary Hermitian LCD codes of any length n ≤ 25 with corresponding dimension k, are presented. In addition, many of these quaternary Hermitian LCD codes given in this paper are optimal and saturate the lower or upper bound of Grassl's code table, and some of them are nearly optimal. [ABSTRACT FROM AUTHOR]

Channel coding technology plays an important role in wireless communication systems, and it serves as a crucial mechanism to reduce interference during the transmission process. As the fifth-generation (5G) and sixth-generation (6G) wireless communication systems rapidly advance, requirements of the users on the quality and security of wireless service are increasing. To solve these problems, it calls for us to explore the new channel coding technologies. In this paper, a linear feedback coding scheme for fading multiple-access channels with degraded message sets (FMAC-DMS) is proposed. In this scheme, the transmitting beamforming and channel splitting are used to transform the channel with complex signals into scalar equivalent sub-channels. Then, the extended Schalkwijk-Kailath coding scheme (SK) is further applied to each sub-channel. The channel capacity, finite blocklength (FBL) sum-rate and FBL secrecy achievable sum-rate of the FMAC-DMS in single-input single-output (SISO) and multi-input single-output (MISO) cases are derived. Finally, we show that the proposed scheme not only provides a FBL coding solution but also guarantees physical layer security(PLS). The numerical and simulation results show the effectiveness of the proposed scheme as a channel coding solution. The study of this paper provides a new method to construct a practical FBL scheme for the FMAC-DMS. [ABSTRACT FROM AUTHOR]

In this paper we consider a family F of 2n-dimensional F q -linear rank metric codes in F q n × n arising from polynomials of the form x q s + δ x q n 2 + s ∈ F q n [ x ] . The family F was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8 , and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8 , providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F q -rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far. [ABSTRACT FROM AUTHOR]

The singleton defect of an [n, k, d] linear code C is defined as s (C) = n - k + 1 - d . Codes with s (C) = s (C ⊥) = 1 are called near maximum distance separable (NMDS) codes. It is known that an [ n , 3 , n - 3 ] NMDS code is equivalent to an (n, 3)-arc in PG(2, q). In this paper, by adding some suitable projective points into some known (q + 5 , 3) -arcs in PG(2, q), we obtain two families of [ q + 7 , 3 , q + 4 ] NMDS codes for even prime power q and a family of [ q + 6 , 3 , q + 3 ] NMDS codes for odd prime power q. In addition, when q = 2 m and m is odd, by adding m suitable projective points into the maximum arcs in PG(2, q), we obtain a family of [ q + m + 2 , 3 , q + m - 1 ] NMDS codes over F q , from which we further induce a family of NMDS codes with parameters [ q t + m + 2 , 3 , q t + m - 1 ] over the extension field F q t for any odd integer t. All the resulting NMDS codes in this paper are shown to be linearly inequivalent to the NMDS codes constructed from elliptic curves, and their weight distributions are completely determined. [ABSTRACT FROM AUTHOR]

Reversible codes have a range of wide applications in cryptography, data storage, and communication systems. In this paper, we investigated reversible codes under the Rosenbloom-Tsfasman metric (RT-metric). First, some properties of reversible codes in the RT-metric were described. An essential condition for a reversible code to be a maximum distance separable code (MDS code, in short) in the RT-metric was established. A necessary condition for a binary self-dual code to be reversible was proven and the same was generalized for q-ary self-dual reversible codes. Several constructions for reversible RT-metric codes were provided in terms of their generator matrices. [ABSTRACT FROM AUTHOR]

In this paper, we use multivariate polynomial rings to construct quantum error-correcting codes (QECCs) via Hermitian construction. We establish a relation between linear codes and ideals of multivariate polynomial rings. We give a necessary and suffcient condition for a multivariate polynomial to generate a Hermitian dual-containing code. By comparing with the literatures in recent years, we construct 31 new QECCs over F q , where q = 3 , 4 , 5 , 7 . Some of them reach quantum singleton bound and some of them exceed quantum GV bound. [ABSTRACT FROM AUTHOR]

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The objective of this paper is to construct some MDS codes with l-Galois hulls of arbitrary dimensions by using the generalized Reed–Solomon codes over finite fields with regard to l-Galois inner product. We give a general construction theorem and some construction ideas of MDS with l-Galois hulls of arbitrary dimensions. Our approach provides a general framework that effectively unifies similar known techniques for constructing MDS codes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method for constructing 1-perfect mixed codes in the Cartesian product F n × F q n , where F n and F q are finite fields of orders n = q m and q. We consider generalized Reed-Muller codes of length n = q m and order (q - 1) m - 2 . Codes whose parameters are the same as the parameters of generalized Reed-Muller codes are called Reed-Muller-like codes. The construction we propose is based on partitions of distance-2 MDS codes into Reed-Muller-like codes of order (q - 1) m - 2 . We construct a set of q q cn nonequivalent 1-perfect mixed codes in the Cartesian product F n × F q n , where the constant c satisfies c < 1 , n = q m and m is a sufficiently large positive integer. We also prove that each 1-perfect mixed code in the Cartesian product F n × F q n corresponds to a certain partition of a distance-2 MDS code into Reed-Muller-like codes of order (q - 1) m - 2 . [ABSTRACT FROM AUTHOR]

Very recently, Heng et al. studied a family of extended primitive cyclic codes. It was shown that the supports of all codewords with any fixed nonzero Hamming weight in this code support a 2-design. In this paper, we study this family of extended primitive cyclic codes in more details. The weight distribution is determined and the parameters of the related 2-designs are also given. Moreover, we prove that the minimum weight codewords in this code support a 3-design when p = 2 , which gives an affirmative answer to Heng's conjecture. [ABSTRACT FROM AUTHOR]

Linear codes play a crucial role in various fields of engineering and mathematics, including data storage, communication, cryptography, and combinatorics. Minimal linear codes, a subset of linear codes, are particularly essential for designing effective secret sharing schemes. In this paper, we introduce several classes of minimal binary linear codes by carefully selecting appropriate Boolean functions. These functions belong to a renowned class of Boolean functions, namely, the general Maiorana–McFarland class. We employ a method first proposed by Ding et al. (IEEE Trans Inf Theory 64(10):6536–6545, 2018) to construct minimal codes violating the Ashikhmin–Barg bound (wide minimal codes) by using Krawtchouk polynomials. The lengths, dimensions, and weight distributions of the obtained codes are determined using the Walsh spectrum distribution of the chosen Boolean functions. Our findings demonstrate that a vast majority of the newly constructed codes are wide minimal. Furthermore, our proposed codes exhibit a significantly larger minimum distance, in some cases, compared to some existing similar constructions. Finally, we address this method, based on Krawtchouk polynomials, more generally, and highlight certain generic properties related to it. These general results offer insights into the scope of this approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a way to partition any constacyclic code over a finite field in its equivalence classes according to the algebraic structure of the code. Such a method gives the generalization of cyclically permutable codes (CPCs), which are called constacyclically permutable codes (CCPCs), and it is useful to derive a CCPC from a given constacyclic code. Moreover, we present an enumerative formula for the code size of such a CCPC, with all of the terms being positive integers, and we provide an algebraic method to produce such a CCPC. [ABSTRACT FROM AUTHOR]

In this paper, we introduced double cyclic codes over Rr × Rs, where R = Z2 + uZ2 = {0, 1, u, 1 + u} is the ring with four elements and u² = 0. We first determined the generator polynomials of R-double cyclic codes for odd integers r and s, then gave the generators of duals of free double cyclic codes over Rr × Rs. By defining a linear Gray map, we looked at the binary images of R-double cyclic codes and gave several examples of optimal parameter binary linear codes obtained from R-double cyclic codes. Moreover, we studied self-dual R-double cyclic codes and presented an example of a self-dual R-double cyclic code. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting (EAQEC) codes are a generalization of quantum error-correcting (QEC) codes, which can be constructed from arbitrary classical linear codes by relaxing the dual-containing condition and by using preshared entangled states between the sender and the receiver. In this paper, we investigate EAQEC codes of length n = 2 (q 2 + 1) a , where q is an odd prime power, a = m 2 + 1 and m is an odd integer. The resulting EAQEC codes are entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes when the minimum distance d ≤ n + 2 2 . [ABSTRACT FROM AUTHOR]

The study of linear codes over local rings, particularly non-chain rings, imposes difficulties that differ from those encountered in codes over chain rings, and this stems from the fact that local non-chain rings are not principal ideal rings. In this paper, we present and successfully establish a new approach for linear codes of any finite length over local rings that are not necessarily chains. The main focus of this study is to produce generating characters, MacWilliams identities and generator matrices for codes over singleton local Frobenius rings of order 32. To do so, we first start by characterizing all singleton local rings of order 32 up to isomorphism. These rings happen to have strong connections to linear binary codes and Z 4 codes, which play a significant role in coding theory. [ABSTRACT FROM AUTHOR]

The hull of a linear code plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. Regarding the quantum error correction, linear codes with determined hull are used to construct quantum codes. In this paper, we focus on the hull of Simplex codes and punctured Simplex codes. We firstly study the properties of the matrix produced by the column vectors of a projective space and determine the Euclidean and Hermitian hull of punctured Simplex codes completely. Secondly, we investigate the Euclidean and Hermitian hull of several classes of linear codes from Simplex codes. [ABSTRACT FROM AUTHOR]

Linear codes of length n over Z p s , p prime, called Z p s -additive codes, can be seen as subgroups of Z p s n . A Z p s -linear generalized Hadamard (GH) code is a GH code over Z p which is the image of a Z p s -additive code under a generalized Gray map. It is known that the dimension of the kernel allows to classify these codes partially and to establish some lower and upper bounds on the number of such codes. Indeed, in this paper, for p ≥ 3 prime, we establish that some Z p s -linear GH codes of length p t having the same dimension of the kernel are equivalent to each other, once t is fixed. This allows us to improve the known upper bounds. Moreover, up to t = 10 if p = 3 or t = 8 if p = 5 , this new upper bound coincides with a known lower bound based on the rank and dimension of the kernel. [ABSTRACT FROM AUTHOR]

In this study, we attempted to demonstrate the interval-valued fuzzy code by extending the concept of an intervalvalued fuzzy set. Further, we discussed the operations of the interval-valued fuzzy code. The interval-valued fuzzy soft code is introduced, and various related properties are investigated in this paper. Finally, we show that the operations of interval-valued fuzzy soft code are discussed. Through this paper, we use the set of integers modulo 2, that is Z2= {0, 1}. [ABSTRACT FROM AUTHOR]

This paper proposes a modification to Tanner's work for constructing girth‐8 quasi‐cyclic low‐density parity‐check codes. The main contribution of this paper is to use an arithmetic sequence at the leftmost column for the exponent matrix so that the lifting size is not necessarily restricted to the prime numbers. Two theorems on the lifting sizes that achieve girth at least 8 using this approach is also provided. This construction exhibits better frame error rate results to the modified 5G new radio (NR) low‐density parity‐check codes for lengths around 500. Also, this construction achieves better frame error rate performance results than the recently proposed one using the Golomb rulers at around frame error rate of 10−6. [ABSTRACT FROM AUTHOR]

Many families of binary nonlinear codes (e.g., Kerdock, Goethals, Delsarte–Goethals, Preparata) can be very simply constructed from linear codes over the Z 4 ring (ring of integers modulo 4), by applying the Gray map to the quaternary symbols. Generalized Kerdock codes represent an extension of classical Kerdock codes to the Z 2 S ring. In this paper, we develop two novel soft-input decoders, designed to exploit the unique structure of these codes. We introduce a novel soft-input ML decoding algorithm and a soft-input soft-output MAP decoding algorithm of generalized Kerdock codes, with a complexity of O (N S log 2 N) , where N is the length of the Z 2 S code, that is, the number of Z 2 S symbols in a codeword. Simulations show that our novel decoders outperform the classical lifting decoder in terms of error rate by some 5 dB. [ABSTRACT FROM AUTHOR]

Let e ≥ 2 and r ≥ 1 be integers, and let R e , r denote the Galois ring of characteristic 2 e and cardinality 2 er. The Teichm u ¨ ller set T r of the Galois ring R e , r can be viewed as the finite field of order 2 r under the addition operation ⊕ and the multiplication operation of R e , r , where for a , b ∈ T r , a ⊕ b is the unique element in T r satisfying a ⊕ b = (a + b) (mod 2). Now a linear code C of length n over T r is said to be k-doubly even if it has a k-dimensional linear subcode C 0 satisfying c · c ≡ 0 (mod 4) for all c ∈ C 0 , where each c ∈ C 0 is viewed as an element of R e , r n and · denotes the Euclidean bilinear form on R e , r n. A k-doubly even code of length n and dimension k over T r is simply called a doubly even code. In this paper, we count all doubly even codes over T r and their two special classes, viz. the codes containing the all-one vector and the codes that do not contain the all-one vector by studying the geometry of a certain special quadratic space over T r. We further provide a recursive method to construct self-orthogonal and self-dual codes of the type { k 1 , k 2 , ... , k e } and length n over R e , r from a (k 1 + k 2 + ⋯ + k e 2 ) -doubly even self-orthogonal code of the same length n and dimension over T r , where n is a positive integer and k 1 , k 2 , ... , k e are non-negative integers satisfying 2 k 1 + 2 k 2 + ⋯ + 2 k e - i + 1 + k e - i + 2 + k e - i + 3 + ⋯ + k i ≤ n for , (here · denotes the floor function and denotes the ceiling function). With the help of this recursive construction method and the enumeration formulae for doubly even codes over T r and their two special classes, we obtain explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over R e , r. Using these enumeration formulae, we classify all self-orthogonal and self-dual codes of lengths 2, 3 and 4 over R 2 , 2 up to monomial equivalence. [ABSTRACT FROM AUTHOR]

M. Fossorier proposed how to determine the necessary and sufficient conditions for the existence of cycles in the Tanner graph of quasi‐cyclic LDPC (QC‐LDPC) codes, which has been widely investigated in the study of LDPC codes. This paper presents some new necessary and sufficient conditions for the existence of cycles with arbitrary lengths and proposes a simple and novel method for counting cycles of QC‐LDPC codes based on the improved conditions. Numerical results show that, compared with the existing methods, the presented method is effective and feasible and can enumerate cycles of QC‐LDPC codes in a cyclic‐shift way. [ABSTRACT FROM AUTHOR]

The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes such as generalized Reed-Solomon codes. In this paper, we define a new extended generalized Reed-Solomon code, i.e., lengthened generalized Reed-Solomon code, which has good algebraic structure and many excellent properties, thus we extend the error-correcting pair to the case for lengthened generalized Reed-Solomon codes. Firstly, we give some sufficient conditions for which an LGRS code is non-GRS, and a necessary and sufficient condition for an LGRS code to be MDS or AMDS, respectively. And then, we constructively determine the existence of the error-correcting pair for lengthened generalized Reed-Solomon codes and give several examples to support our main results. [ABSTRACT FROM AUTHOR]

Binary duadic codes are an interesting subclass of cyclic codes since they have large dimensions and their minimum distances may have a square-root bound. In this paper, we present several families of binary duadic codes of length 2 m - 1 and develop some lower bounds on their minimum distances by using the BCH bound on cyclic codes, which partially solves one case of the open problem proposed in Liu et al (Finite Field Appl 91:102270, 2023). It is shown that the lower bounds on their minimum distances are close to the square root bound. Moreover, the parameters of the dual and extended codes of these binary duadic codes are investigated. [ABSTRACT FROM AUTHOR]

The present paper focuses on investigating the cost-effective configuration of an array of wave energy converters (WECs) composed of vertical cylinders situated in front of a vertical seawall in irregular waves. First, the hydrodynamic calculations are performed using a WAMIT commercial code based on linear potential theory, where the influence of the vertical wall is incorporated using the method of image. The viscous damping experienced by the oscillating cylinder is considered through CFD simulations of a free decay test. A variety of parameters, including WEC diameter, number of WECs, and the spacing between them, are considered to determine an economically efficient WEC configuration. The design of the WEC configuration is aided by a cost indicator, defined as the ratio of the total submerged volume of the WEC to overall power capture. The cost-effective configuration of WECs is achieved when WECs are positioned in front of a vertical wall and the distance between them is kept short. It can be explained that the trapped waves formed between adjacent WECs as well as the standing waves in front of a seawall significantly intensify wave fields around WECs and consequently amplify the heave motion of each WEC. A cost-effective design strategy of WEC deployment enhances the wave energy greatly and, consequently, contributes to constructing the wave energy farm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the general linear group GL(2; 7) of 2×2 invertible matrices over the finite field of order 7 and compute the unit group of the semisimple group alge-bra FqGL(2; 7), where Fq is a finite field. For the computation of the unit group, we need the Wedderburn decomposition of FqGL(2; 7), which is determined by first computing the Wedder-burn decomposition of the group algebra Fq(PSL(3; 2) ... C2). Here PSL(3; 2) is the projective special linear group of degree 3 over a finite field of 2 elements. [ABSTRACT FROM AUTHOR]

This paper provides the results of some computer searches for disjoint maximal (52, 4)-arcs in the known planes of order 16. Thirty-seven new such sets are discovered: four in Johnson plane and thirty-three in Mathon plane, eighteen of which give examples of 104-sets of type (4,8) coming from non-isomorphic pairs of maximal (52, 4)-arcs, providing first examples for such sets. A new lower bound on the number of 104-sets of type (4,8) coming from disjoint maximal (52, 4)-arcs in the known planes of order 16 is obtained. The 104-length binary and ternary linear codes generated by the blocks of 1-designs associated with the known 104-sets of type (4,8) are classified. [ABSTRACT FROM AUTHOR]

Incipient faults in analogue circuits used in complex electrical systems are hard to diagnose due to weak fault features. To improve the reliability and maintainability of analogue circuits in complex electrical systems, a novel incipient soft fault diagnosis method for analogue circuits based on a multilayer dictionary learning and coding network is proposed, including feature preprocessing, linear dictionary feature encoding, and classification modules. In the first module, time–frequency analyses are performed using continuous wavelet transforms to demonstrate the spectrum maps of the fault signals, while scale-invariant feature transforms are used to enhance local features and obtain the keypoint descriptors of the time–frequency spectrum. In the second module, fault features are obtained by locally constrained linear coding (LLC) method using complete dictionaries from the keypoint descriptors acquired in the previous module, which are captured by linear combination of several adjacent atoms in the dictionary learning. To address the limitations of single-layer dictionary learning methods in complete extraction of fault features, a multilayer learning method is used to get richer fault information and improve the diagnosis accuracy. Finally, the linear output features are captured through pooling and fully connected layers. In the third module, the linear features acquired in the second module are quickly classified with simple linear classifiers. The experimental results demonstrate that the proposed method outperforms existing fault diagnosis methods. In order to verify the effectiveness of the proposed method in analogue circuit fault diagnosis, the Sallen–Key filter circuit and four-op-amp biquadratic filter circuit, which are widely used in the field, are selected as experimental circuits in this paper. Specifically, when the component fault values are offset by 20% of their nominal value, the proposed method achieves accuracies of 99.20% for the Sallen–Key bandpass filter circuit and 98.48% for the four-op-amp biquadratic filter circuit. [ABSTRACT FROM AUTHOR]

Self-oscillating systems abound in the natural world and offer substantial potential for applications in controllers, micro-motors, medical equipments, and so on. Currently, numerical methods have been widely utilized for obtaining the characteristics of self-oscillation including amplitude and frequency. However, numerical methods are burdened by intricate computations and limited precision, hindering comprehensive investigations into self-oscillating systems. In this paper, the stability of a liquid crystal elastomer fiber self-oscillating system under a linear temperature field is studied, and analytical solutions for the amplitude and frequency are determined. Initially, we establish the governing equations of self-oscillation, elucidate two motion regimes, and reveal the underlying mechanism. Subsequently, we conduct a stability analysis and employ a multi-scale method to obtain the analytical solutions for the amplitude and frequency. The results show agreement between the multi-scale and numerical methods. This research contributes to the examination of diverse self-oscillating systems and advances the theoretical analysis of self-oscillating systems rooted in active materials. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes a modification to Tanner's work for constructing girth‐8 quasi‐cyclic low‐density parity‐check codes. The main contribution of this paper is to use an arithmetic sequence at the leftmost column for the exponent matrix so that the lifting size is not necessarily restricted to the prime numbers. Two theorems on the lifting sizes that achieve girth at least 8 using this approach is also provided. This construction exhibits better frame error rate results to the modified 5G new radio (NR) low‐density parity‐check codes for lengths around 500. Also, this construction achieves better frame error rate performance results than the recently proposed one using the Golomb rulers at around frame error rate of 10−6.

A linear code C over F q is called Δ -divisible if the Hamming weights wt (c) of all codewords c ∈ C are divisible by Δ . The possible effective lengths of q r -divisible codes have been completely characterized for each prime power q and each non-negative integer r in Kiermaier and Kurz (IEEE Trans Inf Theory 66(7):4051–4060, 2020). The study of Δ -divisible codes was initiated by Harold Ward (Archiv der Mathematik 36(1):485–494, 1981). If t divides Δ but is coprime to q, then each Δ -divisible code C over F q is the t-fold repetition of a Δ / t -divisible code. Here we determine the possible effective lengths of p r -divisible codes over finite fields of characteristic p, where r ∈ N but p r is not a power of the field size, i.e., the missing cases. [ABSTRACT FROM AUTHOR]

Let da(n, 5) and dl(n, 5) be the minimum weights of optimal binary [n, 5] linear codes and linear complementary dual (LCD) codes, respectively. This article aims to investigate dl(n, 5) of some families of binary [n, 5] LCD codes when n = 31s + t ≥ 14 with s integer and t ∈ {2, 8, 10, 12, 14, 16, 18}. By determining the defining vectors of optimal linear codes and discussing their reduced codes, we classify optimal linear codes and calculate their hull dimensions. Thus, the non-existence of these classes of binary [n, 5, da(n, 5)] LCD codes is verified, and we further derive that dl(n, 5) = da(n, 5) − 1 for t, 16 and dl(n, 5) = 16s + 6 = da(n, 5) − 2 for t = 16. Combining them with known results on optimal LCD codes, dl(n, 5) of all [n, 5] LCD codes are completely determined. [ABSTRACT FROM AUTHOR]

In the present work, we classify sets of type (4,n) in PG(3,q). We prove that PG(3,q), apart from the planes of PG(3,3), contains only sets of type (4,n) with standard parameters. Thus, somewhat surprisingly, we conclude that there are no sets of type (4,n) in PG(3,q), q ≠ 3, with non-standard parameters. [ABSTRACT FROM AUTHOR]

Linear codes are the most important family of codes in cryptography and coding theory. Some codes only have a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting p ≡ 1 (mod 4) , we constructed an infinite family of linear codes using two distinct weakly regular unbalanced (and balanced) plateaued functions with index (p − 1) / 2 . Their weight distributions were completely determined by applying exponential sums and Walsh transform. As a result, most of our constructed codes have a few nonzero weights and are minimal. [ABSTRACT FROM AUTHOR]

We introduce a new class of linear codes over a finite field associated to determinantal varieties in the space of hermitian matrices and determine their length, dimension and minimum distance along with the weight spectrum. [ABSTRACT FROM AUTHOR]

The article titled "THREE-WEIGHT AND FIVE-WEIGHT LINEAR CODES OVER FINITE FIELDS" published in the Kragujevac Journal of Mathematics discusses the construction and properties of linear codes derived from defining sets. The authors define a pe-ary linear code C_D using a defining set D and present three-weight and five-weight linear codes along with their weight distributions. They demonstrate that these codes are minimal for m≥5 and can be applied in secret sharing schemes. The article also includes preliminary concepts such as cyclotomic numbers and Gauss sums, which are used in the construction of the linear codes. [Extracted from the article]

It has been known since the 1970's that the difference of the non-zero weights of a projective F q -linear two-weight code has to be a power of the characteristic of the underlying field. Here, we study non-projective two-weight codes and, e.g., show the same result under mild extra conditions. For small dimensions we give exhaustive enumerations of the feasible parameters in the binary case. [ABSTRACT FROM AUTHOR]

We consider the problem of constructing prefix-free codes in which a designated symbol, a space, can only appear at the end of codewords. We provide a linear-time algorithm to construct almost-optimal codes with this property, meaning that their average length differs from the minimum possible by at most one. We obtain our results by uncovering a relation between our class of codes and the class of one-to-one codes. Additionally, we derive upper and lower bounds to the average length of optimal prefix-free codes with a space in terms of the source entropy. [ABSTRACT FROM AUTHOR]

Recently, an infinite class of holographic generalized complexities was proposed. These gravitational observables display the behavior required to be duals of complexity, in particular, linear growth at late times and switchback effect. In this work, we aim to understand generalized complexities in the framework of Lorentzian threads. We reformulate the problem in terms of thread distributions and measures and present a program to calculate the infinite family of codimension-one observables. We also outline a path to understand, using threads, the more subtle case of codimension-zero observables. [ABSTRACT FROM AUTHOR]

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

We present new constructions of binary quantum codes from quaternary linear Hermitian self-dual codes. Our main ingredients for these constructions are nearly self-orthogonal cyclic or duadic codes over F 4 . An infinite family of 0-dimensional binary quantum codes is provided. We give minimum distance lower bounds for our quantum codes in terms of the minimum distance of their ingredient linear codes. We also present new results on the minimum distance of linear cyclic codes using their fixed subcodes. Finally, we list many new record-breaking quantum codes obtained from our constructions. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic density of error-correcting codes with good distance properties and prescribed linearity degree, including (sub)linear and nonlinear codes. We focus on the general setting of finite translation-invariant metric spaces, and then specialize our results to the Hamming metric, to the rank metric, and to the sum-rank metric. Our results show that the asymptotic density of codes heavily depends on the imposed linearity degree and the chosen metric. [ABSTRACT FROM AUTHOR]

Galindo et al. introduced a class of codes which are obtained by evaluation of polynomials at the roots of a trace map (Galindo et al. in IEEE Trans Inform Theory 65: 2593–2602, 2019). Via subfield subcodes, this construction yields new linear codes with good parameters as well as good resulting quantum codes. Here, we extend this construction to allow evaluation at the roots of any polynomial which splits in the field of evaluation. Our proof relies on Galois-closedness of codes in consideration. Moreover, we introduce a lengthening process that preserves Galois-closed property of restricted evaluation codes. Subfield subcodes of such lengthened codes yield further good linear codes. In total, we obtain 17 linear codes over F 4 and F 5 which improve the best known linear code parameters in Grassl (Bounds on the minimum distance of linear codes and quantum codes, 2022, http://www.codetables.de). Moreover, we give a construction for two linear codes which have the best known parameters according to Grassl (Bounds on the minimum distance of linear codes and quantum codes, 2022, http://www.codetables.de), but for which no construction was known before. [ABSTRACT FROM AUTHOR]

This paper introduces a variant of the McEliece cryptosystem and employs the $(C_{1}, C_{1} + C_{2})$ -construction to generate a new code from two arbitrary linear codes. We propose an efficient hard-decision decoding algorithm for linear codes derived from the $(C_{1}, C_{1} + C_{2})$ -construction and integrate them into the McEliece framework. The security of the cryptosystem varies based on the specific codes used in the $(C_{1}, C_{1} + C_{2})$ -construction. Our proposed variant achieves a good level of security with approximately the same key size compared to one of the classic McEliece candidates of the National Institute of Standards and Technology (NIST) standardization process. Specifically, we demonstrate a 25% key size reduction for our proposed parameters compared to one of the 256-bit secured classic McEliece parameters.

In this study, we determine the structure of cyclic codes over the ring Z q [ u ] / 〈 u 2 〉 which is isomorphic to R = Z q + u Z q where q = p s , p is a prime, s is a positive integer, and u 2 = 0. This is equivalent to determining the algebraic structure of ideals of the polynomial quotient ring R [ x ] / 〈 x n − 1 〉 , which is addressed in this paper completely. By establishing the structure of ideals of R [ x ] / 〈 x n − 1 〉 with gcd ⁡ (p , n) = 1 , we present an exact formula that enumerates the number of ideals of this ring that leads to the enumeration of cyclic codes over this ring. Finally, we consider and explore some special families of cyclic codes for some specific q and determine their size. [ABSTRACT FROM AUTHOR]

Remote control over communication networks with bandwidth-constrained channels has attracted considerable recent attention because it holds the promise of enabling a large number of real-time applications, such as autonomous driving, smart grids, and the industrial internet of things (IIoT). However, due to the limited bandwidth, the sub-packets or even bits have to be transmitted successively, thereby experiencing non-negligible latency and inducing serious performance loss in remote control. To overcome this, we introduce an incremental coding method, in which the actuator acts in real time based on a partially received packet instead of waiting until the entire packet is decoded. On this basis, we applied incremental coding to a linear control system to obtain a remote-control scheme. Both its stability conditions and average linear-quadratic-Gaussian-(LQG) cost are presented. Then, we further investigated a multi-user remote-control method, with a particular focus on its applications in the demand response of smart grids over bandwidth-constrained communication networks. The utility loss due to the bandwidth constraint and communication latency are minimized by jointly optimizing the source coding and real-time demand response. The numerical results show that the incremental-coding-aided remote control performed well in both single-user and multi-user scenarios and outperformed the conventional zero-hold control scheme significantly under the LQG metric. [ABSTRACT FROM AUTHOR]

We study a new family of metrics, weighted poset block metric, which generalizes the weighted coordinates poset metric introduced by Panek and Pinheiro (IEEE Trans Inf Theory 66(11):6823–6834, 2020) and the metric for linear error-block codes introduced by Feng et al. (Finite Fields Appl 12(4):638–652, 2006). This metric covers various metrics such as Hamming metric, Lee metric, poset metric, pomset metric, poset block metric, pomset block metric and so on. We give a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product. Moreover, we obtain a Singleton type bound for codes equipped with weighted poset block metric and define MDS codes. When the poset is a chain, we study the relationship between MDS codes and perfect codes. [ABSTRACT FROM AUTHOR]