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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]

Subspace codes, motivated by their extensive application in random network coding, have become one of central topics in algebraic coding theory during the last 10 years. Due to the significant application in subspace codes, Ferrers diagram rank-metric (FDRM) codes also have drawn a lot of attention. In this paper, we introduce two new constructions based on subcodes of MRD codes. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. Apply the first construction to solve the optimality of [ F , 4 ] q -FDRM codes, where F = [ 2 , 2 , 4 , 4 , ... , 2 l , 2 l ] , which was raised in Etzion et al. (IEEE Trans Inf Theory 62:1616–1630, 2016). By the restricted Gabidulin codes and improving the way to select the subcodes, the second construction is presented, which unifies and generalizes all known constructions based on subcodes of Gabidulin codes. By the second construction, we can give new families of optimal FDRM codes, whose numbers of codewords are unequal to q v 0 . This paper also shows new families of FDRM codes whose optimality cannot be obtained by the constructions based on subcodes of F q m -linear MRD codes. [ABSTRACT FROM AUTHOR]

The number-theoretic code is a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts’ codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo $r$. [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]

Linear codes with complementary-duals (LCD codes) are linear codes that trivially intersect with their dual (Massey, 1992). In this paper, we study double circulant codes (DC codes), which are a special class of quasi-cyclic codes, over F4 that are LCD. The main techniques used are as follows: Chinese reminder theory (CRT) decomposition in the line of (Ling et al. 2001), explicit enumeration, and asymptotics. In particular, we show that the class of codes considered here is asymptotically good. [ABSTRACT FROM AUTHOR]

In this paper, several families of irreducible constacyclic codes over finite fields and their duals are studied. The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled. Several families of irreducible constacyclic codes with a few weights and several families of optimal constacyclic codes are constructed. As by-products, a family of [ 2 n , (n - 1) / 2 , d ≥ 2 (n + 1) ] irreducible cyclic codes over GF (q) and a family of [ (q - 1) n , (n - 1) / 2 , d ≥ (q - 1) (n + 1) ] irreducible cyclic codes over GF (q) are presented, where n is a prime such that ord 2 n (q) = (n - 1) / 2 and ord (q - 1) n (q) = (n - 1) / 2 , respectively. The results in this paper complement earlier works on irreducible constacyclic and cyclic codes over finite fields. [ABSTRACT FROM AUTHOR]

Linear codes achieving or nearly achieving the Singleton bound are interesting in both theory and practice. The objective of this paper is to construct several infinite families of MDS, near MDS and almost MDS codes from some special cyclic subgroups of ${\mathbb {F}}_{q^{2}}^{*}$. To this end, the augmentation and extension techniques are used. The codes in this paper have flexible parameters and their lengths could be large. The minimum linear locality of the codes constructed in this paper is also studied. Some infinite families of optimal linearly locally recoverable codes are obtained. Besides, some codes in this paper are proved to be proper for error detection. [ABSTRACT FROM AUTHOR]

In general coding theory, we often assume that error is observed in transferring or storing encoded symbols, while the process of encoding itself is error-free. Motivated by recent applications of coding theory, in this paper, we consider the case where the process of encoding is distributed and prone to error. We introduce the problem of distributed encoding, comprised of a set of $K \in \mathbb {N}$ isolated source nodes and $N \in \mathbb {N}$ encoding nodes. Each source node has one symbol from a finite field, which is sent to each of the encoding nodes. Each encoding node stores an encoded symbol from the same field, as a function of the received symbols. However, some of the source nodes are controlled by the adversary and may send different symbols to different encoding nodes. Depending on the number of the adversarial nodes, denoted by $\beta \in \mathbb {N}$ , and the cardinality of the set of symbols that each one generates, denoted by $v \in \mathbb {N}$ , the process of decoding from the encoded symbols could be impossible. Assume that a decoder connects to an arbitrary subset of $t \in \mathbb {N}$ encoding nodes and wants to decode the symbols of the honest nodes correctly, without necessarily identifying the sets of honest and adversarial nodes. An important characteristic of a distributed encoding system is $t^{*} \in \mathbb {N}$ , the minimum of such $t$ , which is a function of $K$ , $N$ , $\beta $ , and $v$. In this paper, we study the distributed linear encoding system, i.e. one in which the encoding nodes use linear coding. We show that $t^{*}_{\textrm {Linear}}=K+2\beta (v-1)$ , if $N\ge K+2\beta (v-1)$ , and $t^{*}_{\textrm {Linear}}=N$ , if $N\le K+2\beta (v-1)$. In order to achieve $t^{*}_{\textrm {Linear}}$ , we use random linear coding and show that in any feasible solution that the decoder finds, the messages of the honest nodes are decoded correctly. In order to prove the converse of the fundamental limit, we show that when the adversary behaves in a particular way, it can always confuse the decoder between two feasible solutions that differ in the message of at least one honest node. [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]

A functional $k$ -batch code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. Any multiset request of size $k$ of linear combinations (or requests) of the information bits can be recovered by $k$ disjoint subsets of the servers. The goal under this paradigm is to find the minimum number of servers for given values of $s$ and $k$. A recent conjecture states that for any $k=2^{s-1}$ requests the optimal solution requires $2^{s}-1$ servers. This conjecture is verified for $s \leqslant 5$ but previous work could only show that codes with $n=2^{s}-1$ servers can support a solution for $k=2^{s-2} + 2^{s-4} + \left \lfloor{ \frac { 2^{s/2}}{\sqrt {24}} }\right \rfloor $ requests. This paper reduces this gap and shows the existence of codes for $k=\lfloor \frac {5}{6}2^{s-1} \rfloor - s$ requests with the same number of servers. Another construction in the paper provides a code with $n=2^{s+1}-2$ servers and $k=2^{s}$ requests, which is an optimal result. These constructions are mainly based on extended Simplex codes and equivalently provide constructions for parallel Random I/O (RIO) codes. [ABSTRACT FROM AUTHOR]

Let p ≡ 1 (mod 4) be a prime, m a positive integer, ϕ (p m) 2 the multiplicative order of 2 modulo p m , and let q = 2 ϕ (p m) 2 , where ϕ (⋅) is the Euler's function. In this paper, we construct two classes of linear codes over F q and investigate their weight distributions. By calculating two classes of special exponential sums, the desired results are obtained. Let be a prime, a positive integer, the multiplicative order of modulo , and let , where is the Euler's function. In this paper, we construct two classes of linear codes over and investigate their weight distributions. By calculating two classes of special exponential sums, the desired results are obtained. [ABSTRACT FROM AUTHOR]

Negacyclic BCH codes are a subclass of neagcyclic codes and are the best linear codes in many cases. However, there have been very few results on negacyclic BCH codes. Let q be an odd prime power and m be a positive integer. The objective of this paper is to study negacyclic BCH codes with length q m - 1 2 and q m + 1 2 over the finite field GF (q) and analyse their parameters. The negacyclic BCH codes presented in this paper have good parameters in general, and contain many optimal linear codes. For certain q and m, compared with cyclic codes with the same dimension and length, the negacyclic BCH codes presented in this paper have a larger minimum distance in some cases. [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]

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]

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]

DNA code design is a challenging problem, and it has received great attention in the literature due to its applications in DNA data storage, DNA origami, and DNA computing. The primary focus of this paper is in constructing new DNA codes using the cosets of linear codes over the ring Z 4 . The Hamming distance constraint, GC-content constraint, and homopolymers constraint are all considered. In this study, we consider the cosets of Simplex alpha code, Kerdock code, Preparata code, and Hadamard code. New DNA codes of lengths four, eight, sixteen, and thirty-two are constructed using a combination of an algebraic coding approach and a variable neighborhood search approach. In addition, good lower bounds for DNA codes that satisfy important constraints have been successfully established using Magma software V2.24-4 and Python 3.10 programming in our comprehensive methodology. [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]

Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are generalized Reed–Solomon (GRS) codes. The square C 2 of a linear code C is the linear code spanned by the component-wise products of every pair of codewords in C . For an MDS code C , it is convenient to determine whether C is a GRS code by determining the dimension of C 2 . In this paper, we investigate under what conditions that MDS constacyclic codes are GRS. For this purpose, we first study the square of constacyclic codes. Then, we give a sufficient condition that a constacyclic code is GRS. In particular, we provide a necessary and sufficient condition that a constacyclic code of a prime length is GRS. [ABSTRACT FROM AUTHOR]

Hulls of linear codes are widely studied due to their good properties and wide applications. Let n = q m - 1 r and C be an [n, k] cyclic code over F q , where r | q - 1 . In this paper, we present several necessary and sufficient conditions for BCH codes of length n that have k - 1 or k ⊥ - 1 dimensional hulls, where k ⊥ is the dimension of C ⊥ . Further, we give the parameters of several families of self-orthogonal codes that arise as hulls of BCH codes. We obtain many optimal codes. [ABSTRACT FROM AUTHOR]

Consider an assignment of bits to the vertices of a connected graph Γ (V , E) with the property that the value of each vertex is a function of the values of its neighbors. A collection of such assignments is called a storage code of length |V| on Γ . In this paper we construct an infinite family of linear storage codes on triangle-free graphs with rates arbitrarily close to one. [ABSTRACT FROM AUTHOR]

Self-orthogonal codes are linear codes such that they are contained in their duals. Self-orthogonal codes have attracted much attention due to their applications in linear complementary dual codes (LCD codes for short), quantum codes, row-self-orthogonal matrices and so on. In this paper, we first construct several families of ternary self-orthogonal codes from weakly regular bent functions. The parameters and weight distributions of them are determined. Then we use the self-orthogonal codes to construct new infinite families of ternary LCD codes. Some LCD codes are optimal according to the Code Tables at http://www.codetables.de/. [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

A linear code is called a locally repairable code (LRC) with locality r if one can recover an erased code symbol by accessing at most r other code symbols. Constructions of LRCs have been widely investigated in recent years. In this paper, we give a step forward in this direction. Firstly, we propose a novel concept of perfect LRCs whose size exactly achieves the Hamming-type bound, similar to the perfect codes that achieving the Hamming bound in classical coding theory. By the parity-check matrix approach, we establish some important connections between the existence of LRCs and the existence of some subsets of finite geometry and finite fields with certain properties, respectively. By employing q-Steiner systems and sunflowers in projective geometry and difference sets in finite fields, we obtain two new constructions of perfect LRCs with flexible parameters and present several new constructions of k-optimal LRCs achieving another Hamming-type bound under the integers restriction. Moreover, for fixed q and r, the code lengths of all the q-ary r-LRCs constructed in this paper can be arbitrarily large and the code rates can asymptotically achieve the upper bound r r + 1 . [ABSTRACT FROM AUTHOR]

Plateaued functions have been studied in many papers. They can be candidates for designing cryptographic functions and have been used to construct linear codes. They also have close connections to combinatorics and design theory. Plateaued functions on finite abelian groups were studied in Xu (J Comb Des 27:756–783, 2019; J Comb Des 2021, https://doi.org/10.1002/jcd.21281). In this paper, we continue the research in Xu (2019, 2021). We will first study the characterizations of plateaued functions in terms of the derivatives and autocorrelation functions. We will also characterize the plateaued-ness of a function by its distance to affine functions. Then we investigate constructions of plateaued functions. In particular, we will give two general methods to construct plateaued functions and prove the existence of plateaued functions on finite abelian groups whose orders are not prime numbers. We will also show how to construct plateaued functions from a finite abelian group to a group of prime order. As applications of plateaued functions to combinatorics, we will show the existence of two new infinite families of directed strongly regular graphs. [ABSTRACT FROM AUTHOR]

Binary linear codes with few weights have wide applications in communication, secret sharing schemes, authentication codes, association schemes, strongly regular graphs, etc. Projective binary linear codes are among the most important subclasses of binary linear codes for practical applications. In this paper, motivated by the two excellent recent papers (Li et al. in IEEE Trans Inf Theory 67(7):4263–4275, 2021) and (Wang et al. in IEEE Trans Inf Theory 67(8):5133–5148, 2021), several new families of few-weight projective binary linear codes are constructed from the defining sets, and then their Hamming weight distributions are determined by employing the Walsh transform of the corresponding two-to-one functions over finite fields with the even characteristic. Our constructions can produce binary linear codes with new parameters. Some of the constructed binary linear codes are optimal or almost optimal according to the online Database of Grassl, and the duals of some of them are distance-optimal with respect to the sphere packing bound. This paper also shows once again that the two-to-one functions initially studied in (Mesnager and Qu in IEEE Trans Inf Theory 65(12):7884–7895, 2019) are also promising objects in coding theory. Although our derived codes use objects considered in the very recent literature, the analysis of our designed codes involves functions having different algebraic structures (and, therefore, other Walsh transform distribution) and requires solving new systems of equations over finite fields, which is an essential step in determining the weight distribution of our constructed codes. As applications, some of the codes presented in this paper can be used to construct association schemes and secret sharing schemes with interesting access structures. [ABSTRACT FROM AUTHOR]