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In this paper, we employ group rings and skew group rings to construct binary self-dual codes of length 72. There is a well known map σ that sends a (skew) group ring element to an n × n matrix. By using groups of order n, we obtain a lot of generator matrices of the form (I n ∣ σ φ (v) ), where v is an element in a (skew) group ring. Then through special maps, we can send a self-dual code of length 2n over F 2 t to a binary self-dual code of length 2tn. We use these generator matrices to search for binary [72,36,12] self-dual codes and obtain many singly-even and doubly-even codes with new parameters in their weight enumerators that were not known in the literature before. We list our findings on a publicly available website. [ABSTRACT FROM AUTHOR]

We deal with ACD (additive complementary dual) codes and cyclic codes over the mixed alphabet Z 2 R k , where R k : = Z 2 [ y ] / ⟨ y k ⟩ , k ≥ 2 . First, we establish a few criteria for Z 2 R k -additive codes to be ACD codes. We also present conditions for separable codes and a class of additive codes (not necessarily separable) over Z 2 R k to be ACD codes that are both necessary and sufficient. With the help of a Gray map, binary LCD codes are obtained from Z 2 R k -additive codes. Moreover, we describe the generator polynomial of the dual of an additive cyclic code over Z 2 R k . Finally, we construct examples of optimal binary codes as the Gray image of certain additive cyclic codes over Z 2 R k . [ABSTRACT FROM AUTHOR]

In this article we prove Griesmer type bounds for additive codes over finite fields. These new bounds give upper bounds on the length of maximum distance separable (MDS) codes, codes which attain the Singleton bound. We will also consider codes to be MDS if they attain the fractional Singleton bound, due to Huffman. We prove that this bound in the fractional case can be obtained by codes whose length surpasses the length of the longest known codes in the integral case. For small parameters, we provide exhaustive computational results for additive MDS codes, by classifying the corresponding (fractional) subspace-arcs. This includes a complete classification of fractional additive MDS codes of size 243 over the field of order 9. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a one-to-one correspondence between k-tuples of DNA codewords and the elements of a finite group to simulate the reverse and the complement operations of codewords by an automorphism and a translation of the group, respectively, in order to determine the structure of DNA codes. Finally, the case of vector spaces is characterized. [ABSTRACT FROM AUTHOR]

Let a and b be two non-zero elements of a finite field F q , where q > 2 . It has been shown that if a and b have the same multiplicative order in F q , then the families of a-constacyclic and b-constacyclic codes over F q are monomially equivalent. In this paper, we investigate the monomial equivalence of a-constacyclic and b-constacyclic codes when a and b have distinct multiplicative orders. We present novel conditions for establishing monomial equivalence in such constacyclic codes, surpassing previous methods of determining monomially equivalent constacyclic and cyclic codes. As an application, we use these results to search for new linear codes more systematically. In particular, we present more than 70 new record-breaking linear codes over various finite fields, as well as new binary quantum codes. [ABSTRACT FROM AUTHOR]

Linear codes with few weights have many applications in secret sharing, strongly regular graphs, association schemes and authentication codes. Recently, subfield codes of some linear codes with few weights have been studied. In this paper, we present a family of linear codes with few weights and their punctured codes, whose parameters and weight distributions are completely determined. And we give the parameters of their dual codes, which are length-optimal and dimensional-optimal with respect to the Sphere-packing bound. In addition, we also investigate their subfield codes to obtain some q-ary linear codes with few weights. The weight distributions and dualities of these subfield codes are explicitly determined. [ABSTRACT FROM AUTHOR]

The low-hit-zone (LHZ) frequency-hopping sequence (FHS) sets are commonly applied in quasi-synchronous (QS) frequency-hopping multiple access (FHMA) systems to reduce MA interference. In addition, due to the short synchronization time or sophisticated hardware, the correlation window is typically less than the period of the selected FHS set. In this paper, we construct a class of LHZ FHS sets with strictly optimal partial Hamming correlation (PHC) based on the Chinese Remainder Theorem (CRT) and analyze the PHC of our constructed LHZ FHS sets. It turns out that the new LHZ FHS sets are optimal with respect to the Niu-Peng-Fan bound. [ABSTRACT FROM AUTHOR]

Let $ \mathcal {R}_{k}= \mathbb {Z}_4[u_{1},u_{2},\ldots,u_{k}]/\langle u_{i}^2-u_{i},u_{i}u_{j}-u_{j}u_{i}\rangle $ R k = Z 4 [ u 1 , u 2 , ... , u k ] / ⟨ u i 2 − u i , u i u j − u j u i ⟩ be a non-chain ring of characteristic 4, where $ 1\leq i,j\leq k $ 1 ≤ i , j ≤ k and $ k\geq 1 $ k ≥ 1. In this article, we discuss reversible cyclic codes of odd lengths over the ring $ \mathcal {R}_{k} $ R k . We construct bijections between the elements of the ring $ \mathcal {R}_{k} $ R k and DNA- $ 2^{k} $ 2 k bases for k = 1, 2 in such a manner that the reversibility problem is solved. Employing these bijections, reversible complement cyclic codes of odd lengths are generated. Furthermore, we construct a Gray map $ \Phi :\mathcal {R}^{n}_{k}\rightarrow \mathbb {Z}^{2^{k}n}_{4} $ Φ : R k n → Z 4 2 k n and as an application of the Gray map Φ, we obtain the GC-content of cyclic codes of arbitrary odd length over the ring $ \mathcal {R}_{k} $ R k . Meanwhile, we provide some examples of reversible cyclic codes of odd lengths over the ring $ \mathcal {R}_{k} $ R k for different values of k, and also obtain the Lee distances of these codes. [ABSTRACT FROM AUTHOR]

Multi-dimensional cyclic code is a natural generalization of cyclic code. In an earlier paper we explored two-dimensional constacyclic codes over finite fields. Following the same technique, here we characterize the algebraic structure of multi-dimensional constacyclic codes, in particular three-dimensional (α , β , γ) -constacyclic codes of arbitrary length s ℓ k and their duals over a finite field F q , where α , β , γ are non zero elements of F q . We give necessary and sufficient conditions for a three-dimensional (α , β , γ) -constacyclic code to be self-dual. [ABSTRACT FROM AUTHOR]

This study is devoted to the construction of cyclic DNA codes of odd length over Z 4 R . The generator polynomial of cyclic codes of this ring and the structure of its separable codes are studied in detail. A map ψ β from R β to { A , T , G , C } 2 β is defined and then constructed a one-to-one correspondence between DNA codons of alphabets and the elements of R . Furthermore, necessary and sufficient circumstances of cyclic codes over Z 4 , R and Z 4 R to be reversible codes and reverse-complement constraints codes are determined. To support the theories, some important examples are illustrated. [ABSTRACT FROM AUTHOR]

It has been shown that the effectiveness of some algorithms involving permutation code equivalence depends on hulls of linear codes. This paper focuses on studying ternary linear one-dimensional hull codes, and further considers the largest minimum distance d 1 (n , k) among all ternary linear one-dimensional hull [n, k] codes for n ≤ 20 or k ≤ 3 . Most importantly, we classify optimal ternary linear one-dimensional hull [n, 2] codes. Further, we construct some ternary linear one-dimensional hull codes with better minimum distances compared with known results. [ABSTRACT FROM AUTHOR]

Tang and Ding (IEEE Trans Inf Theory 67(1):244–254, 2021) opened a new direction of searching for t-designs from elementary symmetric polynomials, which are used to construct the first infinite family of linear codes supporting 4-designs. In this paper, we first study the properties of elementary symmetric polynomials with 6 or 7 variables over GF (3 m) . Based on them, we present more infinite families of 3-designs that contain some 3-designs with new parameters as checked by Magma for small numbers m. We also construct an infinite family of cyclic codes over GF (q 2) and prove that the codewords of any nonzero weight support a 3-design. In particular, we present an infinite family of 6-dimensional AMDS codes over GF (3 2 m) holding an infinite family of 3-designs and an infinite family of 7-dimensional NMDS codes over GF (3 2 m) holding an infinite family of 3-designs. [ABSTRACT FROM AUTHOR]

This paper focuses on hull dimensional codes obtained by the intersection of linear codes and their dual. These codes were introduced by Assmus and Key and have been the subject of significant theoretical and practical research over the years, gaining increased attention in recent years. Let F q denote the finite field with q elements, and let G be a finite Abelian group of order n. The paper investigates Abelian codes defined as ideals of the group algebra F q G with coefficients in F q . Specifically, it delves into Abelian hull dimensional codes in the group algebra F q G , where G is a finite Abelian group of order n with gcd (n , q) = 1 . Specifically, we first examine general hull Abelian codes and then narrow its focus to Abelian one-dimensional hull codes. Next, we focus on Abelian one-dimensional hull codes and present some necessary and sufficient conditions for characterizing them. Consequently, we generalize a recent result on Abelian codes and show that no binary or ternary Abelian codes with one-dimensional hulls exist. Furthermore, we construct Abelian codes with one-dimensional hulls by generating idempotents, derive optimal ones with one-dimensional hulls, and establish several existing results of Abelian codes with one-dimensional hulls. Finally, we develop enumeration results through a simple formula that counts Abelian codes with one-dimensional hulls in F q G . These achievements exploit the rich algebraic structure of those Abelian codes and enhance and increase our knowledge of them by considering their hull dimensions, reducing the gap between their interests and our understanding of them. [ABSTRACT FROM AUTHOR]

A combinatorial problem concerning the maximum size of the (Hamming) weight set of an [ n , k ] q linear code was recently introduced. Codes attaining the established upper bound are the Maximum Weight Spectrum (MWS) codes. Those [ n , k ] q codes with the same weight set as F q n are called Full Weight Spectrum (FWS) codes. FWS codes are necessarily "short", whereas MWS codes are necessarily "long". For fixed k, q the values of n for which an [ n , k ] q -FWS code exists are completely determined, but the determination of the minimum length M(H, k, q) of an [ n , k ] q -MWS code remains an open problem. The current work broadens discussion first to general coordinate-wise weight functions, and then specifically to the Lee weight and a Manhattan like weight. In the general case we provide bounds on n for which an FWS code exists, and bounds on n for which an MWS code exists. When specializing to the Lee or to the Manhattan setting we are able to completely determine the parameters of FWS codes. As with the Hamming case, we are able to provide an upper bound on M (L , k , q) (the minimum length of Lee MWS codes), and pose the determination of M (L , k , q) as an open problem. On the other hand, with respect to the Manhattan weight we completely determine the parameters of MWS codes. [ABSTRACT FROM AUTHOR]

Let p ≡ 1 (mod 4) be a prime. In this paper, we support a new method, i.e., a product of 2-adic values for four binary sequences, to determine the maximum evaluations of the 2-adic complexity in all almost balanced cyclotomic binary sequences of order four with period N = p , which are viewed as generalizing the results in Hu (IEEE Trans. Inf. Theory 60:5803–5804, 2014) and Xiong et al. (IEEE Trans. Inf. Theory 60:2399–2406, 2014) without the autocorrelation values of cyclotomic binary sequences of order four with period p. By number theory we obtain two necessary and sufficient conditions about the 2-adic complexity of all balanced cyclotomic binary sequences of order four with period N = 2 p and show the 2-adic complexity of several non-balanced cyclotomic binary sequences of order four with period 2p, which are viewed as generalizing the results in Zhang et al. (IEEE Trans. Inf. Theory 66:4613–4620, 2020). [ABSTRACT FROM AUTHOR]

A linear code C of length n = r u + s v is a two-dimensional F -double cyclic code if the set of coordinates can be partitioned into two arrays, such that any cyclic row-shifts and column-shifts of both arrays of a codeword is also a codeword. In this paper, we examine the algebraic structure of these codes and their dual codes in general. Moreover, we are interested in finding out a generating set for these codes (and their dual codes) in case when u = 2 , v = 4 and char (F) ≠ 2 . [ABSTRACT FROM AUTHOR]

In this study, we focus on skew cyclic codes over the family of rings 𝔽q S where q is a power of a prime number and S = 𝔽q + v𝔽q with v2 = v. Structural properties of these codes are studied in detail. Obtained results lead us to characterize 𝔽q S-linear skew cyclic codes. All the minimal spanning sets and generators of these codes are presented. Furthermore, some good 𝔽q -linear codes are obtained as images of the 𝔽q S-linear skew cyclic codes under the Gray mapping.

Let [Formula: see text], where p is an odd prime and r is a positive integer. Consider a ring [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text] with [Formula: see text]. In this paper, we examine the algebraic structure of constacyclic codes over the ring [Formula: see text] of block length [Formula: see text]. Further, a construction of quantum error-correcting codes (QECCs) from constacyclic codes over [Formula: see text] is also given. Moreover, we derive a number of new QECCs.

In this work, we give three new techniques for constructing Hermitian self-dual codes over commutative Frobenius rings with a non-trivial involutory automorphism using λ -circulant matrices. The new constructions are derived as modifications of various well-known circulant constructions of self-dual codes. Applying these constructions together with the building-up construction, we construct many new best known quaternary Hermitian self-dual codes of lengths 26, 32, 36, 38 and 40. [ABSTRACT FROM AUTHOR]

In 2020, Cao et al. proved that any repeated-root constacyclic code is monomially equivalent to a matrix product code of simple-root constacyclic codes. In this paper, we study a family of matrix product codes with wonderful properties, which is a generalization of linear codes obtained from the [ u + v | u - v ] -construction and [ u + v | λ - 1 u - λ - 1 v ] -construction. Then we show that any λ -constacyclic code (not necessary repeated-root λ -constacyclic code) of length N over the finite field F q with gcd (q - 1 ord (λ) , N) ≥ 2 , where ord (λ) is the order of λ in the cyclic group F q ∗ = F q \ { 0 } , is a matrix product code of some constacyclic codes. It is a highly interesting question that the existence of sequences { C 1 , C 2 , C 3 ,... } of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances, i.e., C i is an [ n (C i) , k (C i) , d (C i) ] q -linear code such that lim i → + ∞ n (C i) = + ∞ and lim i → + ∞ d (C i) n (C i) > 0. Based on the [ u + v | λ - 1 u - λ - 1 v ] -construction, we construct several families of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances by using Reed-Muller codes, projective Reed-Muller codes. And we construct some new Euclidean isodual λ -constacyclic codes with square-root-like minimum Hamming distances from Euclidean self-dual cyclic codes and Euclidean self-dual negacyclic codes by monomial equivalences. [ABSTRACT FROM AUTHOR]

Near maximum distance separable (NMDS) codes are important in finite geometry and coding theory. Self-dual codes are closely related to combinatorics, lattice theory, and have important application in cryptography. In this paper, we construct a class of q-ary linear codes and prove that they are either MDS or NMDS which depends on certain zero-sum condition. In the NMDS case, we provide an effective approach to construct NMDS self-dual codes which largely extend known parameters of such codes. In particular, we proved that for square q, almost q/8 NMDS self-dual q-ary codes can be constructed. [ABSTRACT FROM AUTHOR]

In (Shi et al. Finite Fields Appl.80, 102087 2022) studied additive cyclic complementary dual codes with respect to trace Euclidean and trace Hermitian inner products over the finite field F 4 . In this article, we extend their results over F q 2 , where q is an odd prime power. We describe the algebraic structure of additive cyclic codes and obtain the dual of a class of these codes with respect to the trace inner products. We also use generating polynomials to construct several examples of additive cyclic codes over F 9. These codes are better than linear codes of the same length and size. Furthermore, we describe the subfield codes and the trace codes of these codes as linear cyclic codes over F q . [ABSTRACT FROM AUTHOR]

In this paper, we construct a class of Z p r Z p s Z p t -additive cyclic codes generated by 3-tuples of polynomials, where p is a prime number and 1 ≤ r ≤ s ≤ t . We investigate the algebraic structure of these codes and establish that it is possible to determine generator matrices for a subfamily of codes within this class. We employ a probabilistic approach to analyze the asymptotic properties of these codes. For any positive real number δ satisfying 0 < δ < 1 such that the asymptotic Gilbert-Varshamov bound at k + l + n 3 p r - 1 δ is greater than 1 2 , we demonstrate that the relative distance of the random code converges to δ , while the rate of the random code converges to 1 k + l + n . Finally, we conclude that the Z p r Z p s Z p t -additive cyclic codes exhibit asymptotically good properties. [ABSTRACT FROM AUTHOR]

Constacyclic BCH codes are an interesting subclass of constacyclic codes because of their important theoretical and practical value. The purpose of this paper is to study the parameters of cyclic BCH codes of length n = q m - 1 and negacyclic BCH codes of length n = q m - 1 2 . We settle completely their dimensions. We also determine the minimum distances of a class of cyclic BCH codes of length n = q m - 1 and give a lower bound on the minimum distances of other classes of constacyclic BCH codes. As seen by the code examples in this paper, the lower bound on the minimum distances of constacyclic BCH codes we gave is very close to the true minimum distances. These q -ary codes have good parameters in general. [ABSTRACT FROM AUTHOR]

We study the largest minimum weights among quaternary Hermitian LCD codes. We determine the largest minimum weights among quaternary Hermitian LCD codes of length n and dimension k for k ≤ n ≤ 17 . A quaternary Hermitian LCD [21, 5, 13] code and a quaternary Hermitian LCD [21, 9, 9] code are also constructed for the first time. An updated table of the largest minimum weights among quaternary Hermitian LCD [n, k] codes is also given for k ≤ n ≤ 30 . [ABSTRACT FROM AUTHOR]

In this paper, our main objective is to examine the properties and characteristics of 1-generator (2 + u) -quasi-twisted (QT) codes and (2 + u) -generalized quasi-twisted (GQT) codes over the ring Z 4 + u Z 4 , with u 2 = 1 . We determine the structure of the generators and minimal generating sets for both 1-generator (2 + u) -QT and (2 + u) -GQT codes. Additionally, we establish a lower bound for the minimum distance of free 1-generator (2 + u) -QT and (2 + u) -GQT codes over R. Furthermore, we present some numerical examples that illustrate the construction of some optimal Z 4 -linear codes using the Gray map. [ABSTRACT FROM AUTHOR]

In this work, we establish a minimal set of generators for the dual code of a cyclic code having arbitrary length over a finite chain ring. It is observed that this set of generators forms a minimal strong Grobner basis for the dual code. Using this structure for the dual of a cyclic code, we obtain sufficient as well as necessary conditions for a cyclic code to be a self dual code over a finite chain ring. We enumerate the distinct non-trivial self dual cyclic codes over these rings. Further, we determine all MHDR dual cyclic codes. We provide a few examples of self dual and MHDR dual cyclic codes over various finite chain rings. [ABSTRACT FROM AUTHOR]

Subfield codes of linear codes over finite fields have recently attracted great attention due to their wide applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction of 3-dimensional linear codes C f over finite field F 2 m parameterized by any Boolean function f . Then we determine explicitly the weight distributions of C f , the punctured code C ~ f , as well as the corresponding subfield codes over F 2 for several classes of Boolean functions f . In particular, we determine the weight distributions of subfield codes derived from r -plateaued functions. Moreover, the parameters of their dual codes are investigated, which contain length-optimal and dimension-optimal AMDS codes with respect to the sphere packing bound. We emphasize that the new codes are projective and contain binary self-complementary codes. As applications, some of the projective codes we present can be employed to construct s -sum sets for any odd integer s > 1 . [ABSTRACT FROM AUTHOR]

It is very hard to construct an infinite family of cyclic codes of rate close to one half whose minimum distances have a good bound. Tang-Ding codes are very interesting, as their minimum distances have a square-root-like bound. Recently, a new generalization of Tang-Ding codes has been presented, Sun constructed several infinite families of binary cyclic codes with length 2 m - 1 and dimension near 2 m - 1 whose minimum distances much exceed the square-root bound (Sun, Finite Fields Appl. 89, 102200, 2023). In this paper, we construct several families of q-ary cyclic codes with length q m - 1 and dimension near q m - 1 2 , where q ≥ 3 is a prime power and m ≥ 3 is an integer. The minimum distances of these codes and their dual codes much exceed the square-root bound. [ABSTRACT FROM AUTHOR]

We extend a certain type of identities on sums of I-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by I-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of I-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that I-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on Z n whose initial condition is given by a linear code. [ABSTRACT FROM AUTHOR]

Given positive integers k ≤ d and a finite field F , a set S ⊂ F d is (k, c)-subspace evasive if every k-dimensional affine subspace contains at most c elements of S. By a simple averaging argument, the maximum size of a (k, c)-subspace evasive set is at most c | F | d - k . When k and d are fixed, and c is sufficiently large, the matching lower bound Ω (| F | d - k) is proved by Dvir and Lovett. We provide an alternative proof of this result using the random algebraic method. We also prove sharp upper bounds on the size of (k, c)-evasive sets in case d is large, extending results of Ben-Aroya and Shinkar. The existence of optimal evasive sets has several interesting consequences in combinatorial geometry. We show that the minimum number of k-dimensional linear hyperplanes needed to cover the grid [ n ] d ⊂ R d is Ω d (n d (d - k) d - 1 ) , which matches the upper bound proved by Balko et al., and settles a problem proposed by Brass et al. Furthermore, we improve the best known lower bound on the maximum number of incidences between points and hyperplanes in R d assuming their incidence graph avoids the complete bipartite graph K c , c for some large constant c = c (d) . [ABSTRACT FROM AUTHOR]

In this paper, we study linear complementary pairs (LCP) of codes over finite non-commutative local rings. We further provide a necessary and sufficient condition for a pair of codes (C, D) to be LCP of codes over finite non-commutative Frobenius rings. The minimum distances d(C) and d (D ⊥) are defined as the security parameter for an LCP of codes (C, D). It was recently demonstrated that if C and D are both 2-sided LCP of group codes over a finite commutative Frobenius rings, D ⊥ and C are permutation equivalent in Liu and Liu (Des Codes Cryptogr 91:695–708, 2023). As a result, the security parameter for a 2-sided group LCP (C, D) of codes is simply d(C). Towards this, we deliver an elementary proof of the fact that for a linear complementary pair of codes (C, D), where C and D are linear codes over finite non-commutative Frobenius rings, under certain conditions, the dual code D ⊥ is equivalent to C. [ABSTRACT FROM AUTHOR]

For any odd prime p, we provide in this paper two new classes of AMDS symbol-pair cyclic codes of length 4p, one of symbol-pair distance 7 and one of symbol-pair distance 8. This results in two families of AMDS symbol-pair cyclic codes that can be used directly in applications. [ABSTRACT FROM AUTHOR]

BCH codes as a subclass of constacyclic BCH codes have been widely studied, while the results on the parameters of BCH codes over finite fields are still very limited. In this paper, we investigate some q-ary BCH codes and λ -constacyclic BCH codes of length q m + 1 , where q is a prime power and ord (λ) ∣ q - 1 . We determine the dimensions of these codes with some large designed distances, and give good lower bounds on the minimum distance. The code examples presented in this paper indicate that these codes contain many distance-optimal codes and codes with best known parameters. [ABSTRACT FROM AUTHOR]

In this paper, we introduce regular permutation inner products which contain the Euclidean inner product. And we generalize some properties of the Euclidean inner product to regular permutation inner products. As application, we construct a lot of cyclic codes with specific regular permutation hulls and also obtain the dimensions and distances of some BCH codes. [ABSTRACT FROM AUTHOR]

Two skew Hadamard matrices are considered SH-equivalent if they are similar by a signed permutation matrix. This paper determines the number of SH-inequivalent skew Hadamard matrices of order 36 for some types. We also study ternary self-dual codes and association schemes constructed from the skew Hadamard matrices of order 36. [ABSTRACT FROM AUTHOR]

In this article we mainly study linear codes over F 2 n and their binary subfield codes. We construct linear codes over F 2 n whose defining sets are the certain subsets of F 2 n m obtained from mathematical objects called simplicial complexes. We will consider simplicial complexes with one maximal element. The relation of the weights of codewords in two special codes obtained from simplicial complexes is illustrated by using LFSR sequences. And then we determine the parameters of these codes with the help of Boolean functions. As a result, we obtain five infinite families of distance optimal codes and give sufficient conditions for these codes to be minimal. [ABSTRACT FROM AUTHOR]

LCD codes and (almost) optimally extendable codes can be used to safeguard against fault injection attacks (FIA) and side-channel attacks (SCA) in the implementations of block ciphers. The first objective of this paper is to use a family of binary self-orthogonal codes given by Ding and Tang (Cryptogr Commun 12:1011–1033, 2020) to construct a family of binary LCD codes with new parameters. The parameters of the binary LCD codes and their duals are explicitly determined. It turns out that the codes by Ding and Tang are almost optimally extendable codes. The second objective is to prove that two families of known q-ary linear codes given by Heng et al. (IEEE Trans Inf Theory 66(11):6872–6883, 2020) are self-orthogonal. Using these two families of self-orthogonal codes, we construct another two families of q-ary LCD codes. The parameters of the LCD codes are determined and many optimal codes are produced. Besides, the two known families of q-ary linear codes are also proved to be almost optimally extendable codes. [ABSTRACT FROM AUTHOR]

We introduce quaternary modified four μ -circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four μ -circulant Hermitian self-dual codes. We also construct quaternary modified four μ -circulant Hermitian self-dual codes having large minimum weights. Two quaternary Hermitian self-dual [56, 28, 16] codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) [56, 28] codes. In addition, these codes imply the existence of a quantum [[56, 0, 16]] code. [ABSTRACT FROM AUTHOR]

In this work, we propose two new types of codes with locality, namely, locally maximal recoverable (LMR) codes and λ -maximally recoverable (λ -MR) codes. The LMR codes are a subclass of codes with (r , δ) -locality such that they can correct h additional erasures in any one local set, in addition to having (r , δ) -locality. These codes are a restricted case of maximally recoverable (MR) codes, which enable recovery from all information-theoretically correctable erasure patterns in a local set. The λ -MR codes are a subclass of LMR codes which can also handle λ erasures from any coordinate positions. We give constructions for both of these families of codes. We also study the LMR codes that satisfy the complementary dual property. It is well known that codes with this property are capable of safeguarding communication systems against fault injection attacks. We give a construction of distance-optimal cyclic LMR codes that satisfy the complementary dual property. [ABSTRACT FROM AUTHOR]

We investigate CSS and CSS-T quantum error-correcting codes from the point of view of their existence, rarity, and performance. We give a lower bound on the number of pairs of linear codes that give rise to a CSS code with good correction capability, showing that such pairs are easy to produce with a randomized construction. We then prove that CSS-T codes exhibit the opposite behaviour, showing also that, under very natural assumptions, their rate and relative distance cannot be simultaneously large. This partially answers an open question on the feasible parameters of CSS-T codes. We conclude with a simple construction of CSS-T codes from Hermitian curves. The paper also offers a concise introduction to CSS and CSS-T codes from the point of view of classical coding theory. [ABSTRACT FROM AUTHOR]