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In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field 풦 = ℚ(ς8) over ℚ. We employ binary codes to generate a lattice over ℚ(ς8), which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels. [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]

The rank of adjacency matrix plays an important role in construction of linear codes from a directed strongly regular graph using different techniques, namely, code orthogonality, adjacency matrix determinant and adjacency matrix spectrum. The problem of computing the dimensions of such codes is an intriguing one. Several conjectures to determine the rank of adjacency matrix of a DSRG Γ over a finite field, keep researchers working in this area. To address the same to an extent, we have considered the problem of finding the rank over a finite field of the adjacency matrix of a DSRG Γ (v , k , t , λ , μ) with μ = 1 , including some mixed Moore graphs and corresponding codes arising from them, in this paper. [ABSTRACT FROM AUTHOR]

The error-correcting pair is a general algebraic decoding method for linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extensions, we focus on MDS linear codes. Recently, He and Liao showed that for an MDS linear code 풞 with minimum distance 2ℓ + 2, if it has an ℓ-error-correcting pair, then the parameters of the pair have three possibilities. Moreover, for the first case, they gave a necessary condition for an MDS linear code 풞 with minimum distance 2ℓ + 2 to have an ℓ-error-correcting pair, and for the other two cases, they only gave some counterexamples. For the second case, in this paper, we give a necessary condition for an MDS linear code 풞 with minimum distance 2ℓ + 2 to have an ℓ-error-correcting pair, and then basing on the Product Singleton Bound, we prove that there are two cases for such pairs, and then give some counterexamples basing on twisted generalized Reed–Solomon codes for these cases. [ABSTRACT FROM AUTHOR]

In this paper, we study skew constacyclic codes over a class of non-chain rings T = F q [ u 1 , u 2 , ... , u r ] / ⟨ f i (u i) , u i u j - u j u i ⟩ i , j = 1 r , where q = p m , p is some odd prime, m is a positive integer, and f i (u i) are non-constant, monic polynomials that split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over T and their dual. We characterize Euclidean and Hermitian dual-containing skew constacyclic codes. These characterizations serve as a foundational framework for the development of techniques to construct quantum codes. Consequently, we derive plenty of new quantum codes including many maximum distance separable (MDS) quantum codes, and many quantum codes with better parameters than existing ones. Our work further extends to the characterization of skew constacyclic Euclidean and Hermitian linear complementary dual (LCD) codes over T , and we establish that their Gray images also preserve the LCD property. From this analysis, we derive numerous MDS codes and best known linear codes over F q . [ABSTRACT FROM AUTHOR]

Bernal et al. provided a necessary and sufficient condition for a linear code to be realized as an ideal in a finite group algebra and De La Cruz and Willems proved a similar result for ideals in twisted group algebras. In this paper, we extend this characterization to crossed products. Furthermore, we determine conditions for some crossed product codes to be self-dual. [ABSTRACT FROM AUTHOR]

In this paper, we define expanded LRPC codes from LRPC codes with a decoding algorithm using the one of the underlying LRPC codes. Next, we propose to use these codes for cryptography by deriving two cryptosystems in a McEliece setting. in order to reduce the key sizes, we use a generator matrix in systematic form for the first scheme, and an m − order quasi-cyclic LRPC code that we define for the second scheme. The obtained code has a very poor structure and is more likely to be a random linear code. Next, we give some security parameters and compare the key sizes of our public key with the key sizes of some cryptosystems. [ABSTRACT FROM AUTHOR]

Two classes of optimal quaternary sequences have been constructed by applying the sign alternation transform and Gray mapping to Legendre sequence pair, twin-prime sequence pair and GMW sequence pair. In this paper, a new method for investigating the linear complexity over finite field is proposed, and the exact values of the linear complexity over finite field F 4 and Galois ring Z 4 of the quaternary sequences are determined. The results show that their linear complexity are quite good. [ABSTRACT FROM AUTHOR]

Codes over integer rings equipped with the Lee metric are gaining attention among researchers recently. In this paper, we investigate some specific linear codes over integer rings and characterize their dimensions and Lee weights using greatest common divisors (GCD). Furthermore, we construct equidistant Lee codes over ℤ2b, ℤ2pandℤp which have applications in error detection and error correction. [ABSTRACT FROM AUTHOR]

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]

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes (C 1 , C 2) such that C 1 contains C 2 , C 2 is even, and the shortening of the dual of C 1 with respect to the support of each codeword of C 2 is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes (C 1 , C 2) is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature. [ABSTRACT FROM AUTHOR]

There is a non-unital ring I of order 4 defined by generators and relations as I = 〈 a , b | 2 a = 2 b = 0 , a 2 = b , a b = 0 〉. In this paper, we present special constructions of linear codes over I from the adjacency matrices of two class association schemes. These consist of either Strongly Regular Graphs (SRGs) or Doubly Regular Tournaments (DRTs). We investigate the conditions under which these codes are self-orthogonal, quasi self-dual, or Type IV. As a byproduct of this study, some diophantine relations on the weight of sum of vectors with coefficients in I are derived. Some examples of codes with minimum distance better than that of Type IV codes over unital rings of the same order in modest lengths are given. [ABSTRACT FROM AUTHOR]

The construction of (minimal) linear codes from functions has received much attention in the literature. In this paper, we derive several minimal codes from the sets of pre-images of weakly regular plateaued and bent functions over the odd characteristic finite fields. Based on the recent construction method, we obtain six-weight and seven-weight minimal codes from these functions. The Hamming weights in proposed codes are calculated from the character sums of the sets based on the Walsh spectrum of the employed functions, while the weight distributions of the codes are determined from both the sizes of the employed sets and the Walsh distributions of the employed functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the linear subspace of the pseudo-noise (PN)-sequences generated by a primitive polynomial and the simplex code. This family of sequences can be also seen as an Maximum Distance Separable (MDS) 2 -linear code over 2 r . Furthermore, we see how to compute the family of generalized sequences produced by a primitive polynomial by means of a first-order Reed–Muller code. [ABSTRACT FROM AUTHOR]

In this paper, we study linear codes invariant under a cyclic endomorphism T , called T -cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix M , that we call M -cyclic codes, and when M is the companion matrix C f of a given nonzero polynomial f (x) we call them f -cyclic codes. The similarity relation between matrices helps us find connections between M -cyclic codes and f -cyclic codes, where f (x) is the minimal polynomial of M. The class of M -cyclic codes contains cyclic codes and their various generalizations such as constacyclic codes, right and left polycyclic codes, monomial codes, and others. As common in the study of cyclic codes and their generalizations, we make use of the one-to-one correspondence between M -cyclic codes and ideals of the polynomial ring R f : = q [ x ] / 〈 f (x) 〉 , where f (x) is the minimal polynomial of M. This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials among others. Next, we study the duality of these codes, where we show that the b -dual of an M -cyclic code is an M ∗ -cyclic code, where M ∗ is the adjoint matrix of M with respect to b , and we explore some important results on the duality of these codes. Finally, we give examples as applications of some of the results and we construct some optimal codes. [ABSTRACT FROM AUTHOR]

The homogeneous weight (metric) is useful in the construction of codes over a ring of integers Z p l (p prime and l ≥ 1 an integer). It becomes Hamming weight when the ring is taken to be a finite field and becomes Lee weight when the ring is taken to be Z 4 . This paper presents homogeneous weight distribution and total homogeneous weight of burst and repeated burst errors in the code space of n-tuples over Z p l . Necessary and sufficient conditions for existence of an (n, k) linear code over Z p l correcting the error patterns with respect to the homogeneous weight are derived. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Hermitian dual-containing constacyclic codes of length n = q 2 m - 1 q + 1 , where q is a power of 2 and m ≥ 2 is an integer. We apply the Hermitian construction to gain quantum constacyclic codes, and lots of codes obtained in the present paper have parameters better than those of the known quantum codes. The Hermitian hull of a linear code is defined to be the intersection of itself and its Hermitian dual. We construct a family of entanglement-assisted quantum error-correcting (EAQEC) codes with n = 4 m - 1 3 via determining the dimensions of the Hermitian Hulls. [ABSTRACT FROM AUTHOR]

We consider precoder design for maximizing the weighted sum rate (WSR) of successive zero-forcing dirty paper coding (SZF-DPC). For this problem, the existing precoder designs often assume a sum power constraint (SPC) and rely on the singular value decomposition (SVD). The SVD-based designs are known to be optimal but require high complexity. We first propose a low-complexity optimal precoder design for SZF-DPC under SPC, using the QR decomposition. Then, we propose an efficient numerical algorithm to find the optimal precoders subject to per-antenna power constraints (PAPCs). To this end, the precoder design for PAPCs is formulated as an optimization problem with a rank constraint on the covariance matrices. A well-known approach to solve this problem is to relax the rank constraints and solve the relaxed problem. Interestingly, for SZF-DPC, we are able to prove that the rank relaxation is tight. Consequently, the optimal precoder design for PAPCs is computed by solving the relaxed problem, for which we propose a customized interior-point method that exhibits a superlinear convergence rate. Two suboptimal precoder designs are also presented and compared to the optimal ones. We also show that the proposed numerical method is applicable for finding the optimal precoders for block diagonalization scheme. [ABSTRACT FROM PUBLISHER]

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes. In this paper, we elaborate on the implication of these parameters and devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes, our bound is often tight. Furthermore, for these codes, the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights. [ABSTRACT FROM AUTHOR]

An $(r, \delta)$ -locally repairable code ($(r, \delta)$ -LRC for short) was introduced by Prakash et al. for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$ -LRCs produced by Gopalan et al.. An $(r, \delta)$ -LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. generalized the construction of cyclic $r$ -LRCs proposed by Tamo et al. , and constructed several classes of optimal $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of , , this paper first characterizes $(r, \delta)$ -locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively from the product of two sets of zeros. Our constructions include all optimal cyclic $(r,\delta)$ -LRCs proposed in , , and our method seems more convenient to obtain optimal cyclic $(r, \delta)$ -LRCs with flexible parameters. Moreover, many optimal cyclic $(r,\delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively with $(r+\delta -1)\nmid n$ can be obtained from our method. [ABSTRACT FROM AUTHOR]

In this paper, we show that the communication channel known as writing in memory with defective cells is a relevant information-theoretical model for a specific case of passive warden steganography when the sender embeds a secret message into a subset C of the cover object X without sharing C with the recipient. The set C, which is also called the selection channel, could be arbitrary, determined by the sender from the cover object using a deterministic, pseudo-random, or a truly random process. We call this steganography "writing on wet paper" and realize it using a simple variable-rate random linear code that gives the sender a convenient flexibility and control over the embedding process and is thus suitable for practical implementation. The importance of the wet paper scenario for covert communication is discussed within the context of adaptive steganography and perturbed quantization steganography. Heuristic arguments supported by tests using blind steganalysis indicate that the wet paper steganography provides improved steganographic security and is less vulnerable to steganalytic attacks compared with existing methods with shared selection channels. [ABSTRACT FROM PUBLISHER]

Let p be an odd prime with p ≠ 5 . In this paper, we first provide the structures of repeated-root cyclic codes of length 10 p s over finite fields F p m . We then give two methods of constructing good quantum error-correcting (QEC) codes from repeated-root cyclic codes of length 10 p s over finite fields F p m . By means of the dimensions of repeated-root cyclic codes of length 10 p s over finite fields F p m , we exhibit an effective manner for constructing new EAQEC codes. We show that the usage of these methods brings us many good QEC and EQAEC codes having these advantages: (1) the parameters of our QEC and EQAEC codes are different from all the previous constructions; (2) for repeated-root cyclic codes, our methods allows for easily calculating the dimensions of QEC and EQAEC codes, and the numbers c of pre-shared maximally entangled states of EAQEC codes. [ABSTRACT FROM AUTHOR]

Applied in communication, data storage system, secret sharing schemes, authentication codes and association schemes, linear codes attract much attention. In this paper, a class of three-weight linear codes is obtained by the defining sets over finite fields of odd characteristic. The parameters and weight distributions of linear codes are determined by the additive characters, multiplicative characters and Gauss sums. Further, most of linear codes obtained are minimal, which can be used to construct secret sharing schemes. [ABSTRACT FROM AUTHOR]

Let I be the commutative non-unital ring of order 4 defined by generators and relations. I = a , b ∣ 2 a = 2 b = 0 , a 2 = b , a b = 0. Alahmadi et al. have classified QSD codes, Type IV codes (QSD codes with even weights) and quasi-Type IV codes (QSD codes with even torsion code) over I up to lengths n = 6 , and suggested two building-up methods for constructing QSD codes. In this paper, we construct more QSD codes, Type IV codes and quasi-Type IV codes for lengths n = 7 and 8, and describe five new variants of the two building-up construction methods. We find that when n = 8 there is at least one QSD code with minimun distance 4, which attains the highest minimum distance so far, and we give a generator matrix for the code. We also describe some QSD codes, Type IV codes and quasi-Type IV codes with new weight distributions. [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]

By generalizing the stabilizer quantum error-correcting codes, entanglement-assisted quantum error-correcting (EAQEC) codes were introduced, which could be derived from any classical linear codes via the relaxation of self-orthogonality conditions with the aid of pre-shared entanglement between the sender and the receiver. In this paper, three classes of entanglement-assisted quantum error-correcting maximum-distance-separable (EAQMDS) codes are constructed through generalized Reed–Solomon codes. Under our constructions, the minimum distances of our EAQMDS codes are much larger than those of the known EAQMDS codes of the same lengths that consume the same number of bits. Furthermore, some of the lengths of the EAQMDS codes are not divisors of q 2 - 1 , which are completely new and unlike all those known lengths existed before. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in right (resp., left) quasi-polycyclic (QP) codes of length n = m ℓ with an associated vector a = (a 0 , a 1 , ... , a m - 1) ∈ F q m , which are a generalization of quasi-cyclic codes (QC) and quasi-twisted (QT) codes. They are defined as invariant subspaces of F q n by the right (resp., left) QP operator T ~ a → (resp., T ~ a ← ). A correspondence between the right (resp., left) ℓ -QP codes and the linear codes of length ℓ over the ring R a → , m : = F q [ x ] / x m - a → (x) , a → (x) = ∑ i = 0 m - 1 a i x i (resp., R a ← , m : = F q [ x ] / x m - a ← (x) , a ← (x) = ∑ i = 0 n - 1 a i x m - 1 - i ) is given. This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials. Moreover, we also discuss the structure of the 1-generator QP codes, and we prove BCH-like and the Hartmann–Tzeng-like bounds on the minimum distance of QP codes. Finally, we give examples of new quantum codes derived from QP codes as an application of some of the results. Several of these codes are MDS. [ABSTRACT FROM AUTHOR]

The dimension of Euclidean hull for a linear code is an important quantity to determine the parameters of an entanglement-assisted quantum error-correcting (EAQEC, for short) code. In this paper, we study the Euclidean hull of a cyclic code over finite non-chain rings Z 4 + v Z 4 , where v 2 = v . We provide a method of constructing EAQEC codes by means of our results. This construction is applied to obtain numerous new EAQEC codes, and some of them have better parameters than current EAQEC codes available. [ABSTRACT FROM AUTHOR]

In this paper, we introduce F q 2 -double cyclic codes of length n = r + s , where F q 2 is the Galois field of q 2 elements, q is a power of a prime integer p and r, s are positive integers. We determine the generator polynomials for any F q 2 -double cyclic code. For any F q 2 -double cyclic code C , we will define the Euclidean dual code C ⊥ based on the Euclidean inner product and the Hermitian dual code C ⊥ H based on the Hermitian inner product. We will construct a relationship between C ⊥ and C ⊥ H and then find the generator polynomials for the Hermitian dual code C ⊥ H. As an application of our work, we will present examples of optimal parameter linear codes over the finite field F 4 and also examples of optimal quantum codes that were derived from F 4 -double cyclic codes using the Hermitian inner product. [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]

Linear codes with complementary duals (shortly named LCD codes) are linear codes whose intersection with their duals are trivial. In this paper, we give a method of constructing these type of linear codes from equitable partitions of association schemes. The LCD codes constructed in this paper are of length 2n and dimension n and have the property of being formally self-dual. To illustrate the method we construct LCD codes from some distance-regular graphs. [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]

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

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]

The primary application of codes with locality and availability in distributed storage is for data recovery in case that data are lost on some damaged servers. Locality enables the recovery of lost data by contacting only a restricted number of remaining servers. Availability means that more than one subset of servers providing locality is available for data recovery for each server. The secondary application of these codes in distributed storage is to allow access to hot data in times of high demand. In this paper it is shown that the binary linear [14, 4, 7] code has locality 2 and availability 6 and it can be interpreted as a three-dimensional graph obtained from a [7, 3, 4] Simplex code. It is achieving upper bounds on basic parameters for codes with all-symbols locality and availability. This code can be a building element of more complex codes with scalability inspired by three-dimensional structures. The availability spectrum is introduced as a tool for analyzing codes with locality and availability. [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]

Let R v , q = F q + v F q with v 2 = v . In this paper, we provide three methods of constructing quantum synchronizable codes (QSCs) by using the Euclidean sums of (1 - 2 v) -constacyclic codes over R v , q . Concrete examples are presented to enrich the variety of available QSCs. In addition, two of methods in our research are easier than available methods for constructing QSCs and producing more QSCs. [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]

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]

In this paper, we are interested in linear codes invariant under the (σ , δ) -action of a given matrix M , that we call (M , σ , δ) -skew codes. Various subclasses of these codes, depending on the (σ , δ) -cyclical property of M , are derived. In particular, we distinguish between three fundamental types: (M , σ , δ) -cyclic, (M , σ , δ) -quasi-cyclic and (M , σ , δ) -multi-cyclic. These different subclasses allow us to obtain some of the well known types of linear codes as special cases depending on the choice of M. We observe that the (σ , δ) -similarity in M n (F q ) gives rise to various relations between these codes. As common in the study of cyclic codes and their generalizations, we use the skew polynomial ring F q [ x , σ , δ ] to study (M , σ , δ) -skew codes. As an application of the theory developed here, we construct some optimal linear codes and MDS codes over different alphabets with additional desirable properties such as self orthogonality and linear complementary duality (LCD). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the structure and properties of skew negacyclic codes and skew quasi-negacyclic codes over the ring R = ℤ q + u ℤ q , u 2 = 0. Some structural properties of R [ x , Θ ] are discussed, where Θ is an automorphism of R. A skew quasi-negacyclic code of length s ℓ with index ℓ over R is viewed both as in the conventional row circulant form and also as an R [ x , Θ ] / (x s + 1) -submodule of G R (R , ℓ) [ x , δ ] / (x s + 1) , where G R (R , ℓ) is the Galois extension ring of degree ℓ over R and δ is an automorphism of G R (R , ℓ). A sufficient condition for one generator skew quasi-negacyclic codes to be free is determined. Some distance bounds for free one generator skew quasi-negacyclic codes are discussed. Furthermore, given the decomposition of a skew quasi-negacyclic code, we provide the decomposition of its dual code. As a result, a characterization of self-dual skew quasi-negacyclic codes over R = ℤ q + u ℤ q is provided. By using computer search we obtained a number of new linear codes over ℤ 4 from skew negacyclic and skew quasi-negacyclic codes over ℤ 4 + u ℤ 4 . [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]

The residual belief propagation (RBP) algorithm, which is the most classic informed dynamic scheduling strategy, achieves outstanding performance in error correction and can drastically accelerate convergence speed. However, the greedy algorithmic property of this iterative decoding will inevitably cause loss of decoding performance. To address this, a novel algorithm called the partial average bundle residual belief propagation (PABRBP) is proposed in this paper. According to the construction characteristics of a base matrix of protograph‐LDPC codes, informed dynamic scheduling (IDS) strategies are applied to an edge bundle of base matrices for the first time. This edge bundle of the base matrix can be applied to a corresponding cyclic permutation matrix. Furthermore, the update level of each bundle is determined by the value of the Partially Average Bundle Residual (PABR). Therefore, the edge message with the maximum residual in each bundle is updated in order, and the process of iterative decoding is less likely to become trapped in a local optimum. Additionally, the generation of silent nodes is reduced as much as possible. To further improve the PABRBP decoding performance for medium and long codes over the fading channel, the adjusted compensation term is periodically introduced. Analysis and simulation results show that PABRBP demonstrates a notable convergence quality and decoding performance improvement over the fading channels compared to existing state‐of‐art IDS algorithms. [ABSTRACT FROM AUTHOR]

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

Hermitian dual-containing codes are an important class of linear codes which have important applications in the construction of quantum codes. In this paper, two classes of Hermitian dual-containing almost MDS codes over finite fields are studied. By employing the Hermitian construction, a class of quantum codes with minimum distance 3 and a class of quantum codes with minimum distance 4 are constructed. [ABSTRACT FROM AUTHOR]

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

A class of optimal three-weight [ q k - 1 , k + 1 , q k - 1 (q - 1) - 1 ] cyclic codes over I F q , with k ≥ 2 , achieving the Griesmer bound, was presented by Heng and Yue (IEEE Trans Inf Theory 62(8):4501–4513, 2016. https://doi.org/10.1109/TIT.2016.2550029). In this paper we study some of the subfield codes of this class of optimal cyclic codes when k = 2 . The weight distributions of the subfield codes are settled. It turns out that some of these codes are optimal and others have the best known parameters. The duals of the subfield codes are also investigated and found to be almost optimal with respect to the sphere-packing bound. In addition, the covering structure for the studied subfield codes is determined. Some of these codes are found to have the important property that any nonzero codeword is minimal, which is a desirable property that is useful in the design of a secret sharing scheme based on a linear code. Moreover, a specific example of a secret sharing scheme based on one of these subfield codes is given. Finally, a class of optimal two-weight linear codes over I F q , achieving the Griesmer bound, whose duals are almost optimal with respect to the sphere-packing bound is presented. Through a different approach, this class of optimal two-weight linear codes was reported very recently by Heng (IEEE Trans Inf Theory 69(2):978–994, 2023. https://doi.org/10.1109/TIT.2022.3203380). Furthermore, it is shown that these optimal codes can be used to construct strongly regular graphs. [ABSTRACT FROM AUTHOR]