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In this paper, we introduce several new construction techniques of linear complimentary dual (LCD) codes. First, we show that if a LCD code is fixed by a transitive automorphism group, then the punctured code is also LCD. We show that several important families such as anti-primitive BCH codes and extended Gabidulin codes are LCD. Finally relationships among self-dual codes, λ-orthogonal matrices and LCD codes are studied. As an application we give LCD codes from Paley-type bipartite graph and from punctured MacDonalds codes. Further, we give a bound on some LCD codes. [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]

In this paper, we not only give the parity check matrix of the [1, 0]-twisted generalized Reed-Solomon (in short, TGRS) code, but also determine the weight distribution. Especially, we show that the [1, 0]-TGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the [1, 0]-TGRS code to be self-orthogonal, and then construct several classes of self-dual or almost self-dual [1, 0]-TGRS codes. Finally, on the basis of these self-dual or almost self-dual [1, 0]-TGRS codes, we obtain some LCD [1, 0]-TGRS codes. [ABSTRACT FROM AUTHOR]

Let p be a prime number and q = p m for some positive integer m. In this paper, we find the possible Hermitian hull dimensions of λ -constacyclic codes over R e = F q 2 + u F q 2 + u 2 F q 2 + ⋯ + u e - 1 F q 2 , u e = 1 where F q 2 is the finite field of q 2 elements, e | (q + 1) and λ = η 1 α 1 + η 2 α 2 + ⋯ + η e α e for α l ∈ F q 2 ∗ of order r l such that r l ∣ q + 1 (for each 1 ≤ l ≤ e ). Further, we obtain some conditions for these codes to be Hermitian LCD. Also, under certain conditions, we establish a strong result that converts every constacyclic code to a Hermitian LCD code (Corollaries 2 and 3). We also study the structure of generator polynomials for Hermitian dual-containing constacyclic codes (Theorems 8 and 9), and obtain parameters of quantum codes using the Hermitian construction. The approach we used to derive Hermitian dual-containing conditions via the hull has not been used earlier. As an application, we obtain several optimal and near-to-optimal LCD codes, constacyclic codes having small hull dimensions, and quantum codes. [ABSTRACT FROM AUTHOR]

In this paper, we find a necessary and sufficient condition for multi-twisted Reed-Solomon codes to be MDS. In particular, we introduce a new class of MDS double-twisted Reed-Solomon codes C α , t , h , η with twists t = (1 , 2) and hooks h = (0 , 1) over the finite field F q , providing a non-trivial example over F 16 and enumeration over the finite fields of size up to 17. Moreover, we obtain necessary conditions for the existence of multi-twisted Reed-Solomon codes with small dimensional hull. Consequently, we derive conditions for the existence of MDS multi-twisted Reed-Solomon codes with small dimensional hull. [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 $ \mathbb{F}_4 $ 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.

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]

Let GF(l) be the Galois field with l = pm elements where p is a prime number and integer m ≥ 1. Here, we present three constructions for linear codes over GF(l) (depending on the parity of l) by using the quadratic residue approach and obtain some sufficient conditions for these codes to be LCD with respect to the Euclidean and Hermitian inner products, respectively. Furthermore, several examples of codes, including optimal and near to optimal codes, are provided to support our study. [ABSTRACT FROM AUTHOR]

We examine the binary codes from the adjacency matrices of various products of graphs, and show that if the binary codes of a set of graphs have the property that their dual codes are the codes of the associated reflexive graphs, and are thus LCD, i.e. have zero hull, then, with some restrictions, the binary code of the product will have the same property. The codes are candidates for decoding using this property, or also, in the case of the direct product, by permutation decoding. [ABSTRACT FROM AUTHOR]

In this paper, we investigate self-dual double circulant, and self-dual and linear complementary dual (LCD) double negacirculant codes over a finite ring $R = \mathbb F_{q} + u \mathbb F_{q} + v \mathbb F_{q} + uv\mathbb F_{q}$ , where $u^{2}=u$ , $v^{2}=v$ , $uv=vu $ and $q=p^{m}$ . We study the algebraic structure of double circulant codes over $R$ . We provide necessary and sufficient conditions for a double circulant code to be a self-dual code. We give a formula to get the total number of self-dual double circulant codes over the ring $R$ . We compute distance bounds for self-dual double circulant codes over $R$ . In addition, by using a Gray map, we show that the families of self-dual double circulant codes under the Gray map are asymptotically good. Moreover, the algebraic structure of double negacirculant codes and necessary and sufficient conditions for a double negacirculant code to be a self-dual code and to be an LCD code are also given. We determine the total number of self-dual and LCD double negacirculant codes over $R$ .

In this paper, we present a basic theory of the duality of linear codes over three of the non-unital rings of order four; namely $I$ , $E$ , and $H$ as denoted in (Fine, 1993). A new notion of duality is introduced in the case of the non-commutative ring $E$ . The notion of self-dual codes with respect to this duality coincides with that of quasi self-dual codes over $E$ as introduced in (Alahmadi et al, 2022). We characterize self-dual codes and LCD codes over the three rings, and investigate the properties of their corresponding additive codes over $\mathop {\mathrm {\mathbb {F}}} _{4}$ . We study the connection between the dual of any linear code over these rings and the dual of its associated binary codes. A MacWilliams formula is established for linear codes over $E$ .

In this paper, we discuss double circulant and double negacirculant codes over a family of finite rings R t = F q + v 1 F q + v 2 F q + ⋯ + v t F q ; (v i 2 = v i , v i v j = v j v i = 0 , i , j = 1 , 2 , ... , t , i ≠ j) , where q is an odd prime power. We obtain necessary and sufficient conditions for double circulant codes (double negacirculant codes) to be self-dual codes and to be linear codes with complementary dual (or LCD codes) codes and study the algebraic structure of self-dual and LCD double circulant (double negacirculant codes) codes. We derive a formula to determine the total number of self-dual and LCD double circulant codes (double negacirculant codes) over the ring Rt. We also find distance bounds for double circulant codes over Rt. Moreover, we use a Gray map to prove that the families of self-dual and LCD double circulant codes under this Gray map are asymptotically good. [ABSTRACT FROM AUTHOR]

In this work, we study cyclic codes of length n over a finite commutative non-chain ring ℛ = F q [ u , v ] / 〈 u 2 − γ u , v 2 − ϵ v , u v − v u 〉 where γ , ϵ ∈ F q * and we find new and better quantum error-correcting codes than previously known quantum error correcting codes. Then certain constraints are imposed on the generator polynomials of cyclic codes, so these codes become linear complementary dual codes (in short LCD codes). We then verify that the Gray image of linear complementary dual codes of length n over ℛ is a linear complementary dual code of length 4 n over F q by establishing a Gray map. [ABSTRACT FROM AUTHOR]

The hull of a linear code C is the intersection of C with its dual. To the best of our knowledge, there are very few constructions of binary linear codes with the hull dimension ≥ 2 except for self-orthogonal codes. We propose a building-up construction to obtain a plenty of binary [ n + 2 , k + 1 ] codes with hull dimension ℓ , ℓ + 1 , or ℓ + 2 from a given binary [n, k] code with hull dimension ℓ . In particular, with respect to hull dimensions 1 and 2, we construct all binary optimal [n, k] codes of lengths up to 13. With respect to hull dimensions 3, 4, and 5, we construct all binary optimal [n, k] codes of lengths up to 12 and the best possible minimum distances of [13, k] codes for 3 ≤ k ≤ 10 . As an application, we apply our binary optimal codes with a given hull dimension to construct several entanglement-assisted quantum error-correcting codes (EAQECCs) with the best known parameters. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the algebraic structure of the non-local ring R q = F q [ v ] / 〈 v 2 + 1 〉 and identify the automorphisms of this ring to study the algebraic structure of the skew constacyclic codes and their duals over this ring. Furthermore, we give a necessary and sufficient condition for the skew constacyclic codes over R q to be linear complementary dual (LCD). We present some examples of Euclidean LCD codes over R q and tabulate the parameters of Euclidean LCD codes over finite fields as the Φ -images of these codes over R q , which are almost maximum distance separable (MDS) and near MDS. Eventually, by making use of Hermitian linear complementary duals of skew constacyclic codes over R q and the map Φ , we give a class of entanglement-assisted quantum error correcting codes (EAQECCs) with maximal entanglement and tabulate parameters of some EAQECCs with maximal entanglement over finite fields. [ABSTRACT FROM AUTHOR]

Recently, a new algorithm to test equivalence of two cyclic codes has been introduced which is efficient and produced useful results. In this work, we generalize this algorithm to constacyclic codes. As an application of the algorithm we found many constacyclic codes with good parameters and properties. In particular, we found 22 new codes that improve the minimum distances of best known linear codes (BKLCs). [ABSTRACT FROM AUTHOR]

One of the most important and challenging problems in coding theory is to construct codes with best possible parameters and properties. The class of quasi-cyclic (QC) codes is known to be fertile to produce such codes. Focusing on QC codes over the binary field, we have found 113 binary QC codes that are new among the class of QC codes using an implementation of a fast cyclic partitioning algorithm and the highly effective ASR algorithm. Moreover, these codes have the following additional properties: a) they have the same parameters as best known linear codes, and b) many of the have additional desired properties such as being reversible, LCD, self-orthogonal or dual-containing. Additionally, we present an algorithm for the generation of new codes from QC codes using ConstructionX, and introduce 33 new record breaking linear codes over GF(2), GF(3) and GF(5) produced from this method. [ABSTRACT FROM AUTHOR]

This paper presents a formula for the dimension of Galois hulls of constacyclic codes. For this, we have arranged the irreducible factors of xn − λ over the finite field F q in a suitable way. Also, considering some restrictions on q, the number of constacyclic codes of length n over F q is calculated for a given Galois hull dimension. [ABSTRACT FROM AUTHOR]

For m ≥ 4 even, the duals of p-ary codes, for any prime p, from adjacency matrices for the m-ary 2-cube Q 2 m are shown to have subcodes with parameters [ m 2 , 2 m - 2 , m ] for which minimal PD-sets of size m 2 are constructed, hence attaining the full error-correction capabilities of the code, and, as such, the most efficient sets for full permutation decoding. [ABSTRACT FROM AUTHOR]

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull (LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasi-abelian, and sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed linear codes. As an application, using these conditions, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) with respect to the online Database of Grassl. [ABSTRACT FROM AUTHOR]

Linear complementary dual (abbreviated LCD) generalized Gabidulin codes (including Gabidulin codes) have been recently investigated by Shi and Liu et al. (Shi et al. IEICE Trans. Fundamentals E101-A(9):1599-1602, 2018, Liu et al. Journal of Applied Mathematics and Computing 61(1): 281-295, 2019). They have constructed LCD generalized Gabidulin codes of length $ n $ over $ \mathbb{F}_{q^{n}} $ by using self-dual bases of $ \mathbb{F}_{q^{n}} $ over $ \mathbb{F}_{q} $ when $ q $ is even or both $ q $ and $ n $ are odd. Whereas for the case of odd $ q $ and even $ n $, whether LCD generalized Gabidulin codes of length $ n $ over $ \mathbb{F}_{q^{n}} $ exist or not is still open. In this paper, it is shown that one can always construct LCD generalized Gabidulin codes of length $ n $ over $ \mathbb{F}_{q^{n}} $ for the case of odd $ q $ and even $ n $.

Additive codes have become an increasingly important topic in algebraic coding theory due to their applications in quantum error-correction and quantum computing. Linear Complementary Dual (LCD) codes play an important role for improving the security of information against certain attacks. Motivated by these facts, we define additive complementary dual codes (ACD for short) over a finite abelian group in terms of an arbitrary duality on the ambient space and examine their properties. We show that the best minimum weight of ACD codes is always greater than or equal to the best minimum weight of LCD codes of the same size and that this inequality is often strict. We give some matrix constructions for quaternary ACD codes from a given quaternary ACD code and also from a given binary self-orthogonal code. Moreover, we construct an algorithm to determine if a given quaternary additive code is an ACD code with respect to all possible symmetric dualities. We also determine the largest minimum distance of quaternary ACD codes for lengths $n \leq 10$. The obtained codes are either optimal or near optimal according to Bierbrauer et al +. (2009). [ABSTRACT FROM AUTHOR]

We determine the unique factorization of some polynomials over a finite local commutative ring with identity explicitly. This solves and generalizes the main conjecture of Qian, Shi and Solé in [13]. We also give some applications to enumeration of certain generalized double circulant self-dual and linear complementary dual (LCD) codes over some finite rings together with an application in asymptotic coding theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the algebraic structure of the non-local ring Rq=Fq[v]/⟨v2+1⟩ and identify the automorphisms of this ring to study the algebraic structure of the skew constacyclic codes and their duals over this ring. Furthermore, we give a necessary and sufficient condition for the skew constacyclic codes over Rq to be linear complementary dual (LCD). We present some examples of Euclidean LCD codes over Rq and tabulate the parameters of Euclidean LCD codes over finite fields as the Φ-images of these codes over Rq, which are almost maximum distance separable (MDS) and near MDS. Eventually, by making use of Hermitian linear complementary duals of skew constacyclic codes over Rq and the map Φ, we give a class of entanglement-assisted quantum error correcting codes (EAQECCs) with maximal entanglement and tabulate parameters of some EAQECCs with maximal entanglement over finite fields.

An MRD code (n , k , n − k + 1) is a k -dimensional q N -linear subspace of the n -dimensional vector space q N n over q N for n ≤ N. Linear codes ′ (n ′ , k ′ , d) (of length n ′ ≥ n , dimension k ′ ≥ k) that are obtained from MRD codes (n , k , d) satisfying the property ′ ∩ ′ ⊥ = { 0 } are defined to be LCD MRD codes. An inherent relationship between the generator G and parity-check matrices H of LCD MRD codes is observed. This in fact identifies LCD MRD codes as trivial (G G T = I) and nontrivial (G G T ≠ I) codes. Further, two classes of LCD MRD codes of length n ℓ are constructed from ℓ (trivial and nontrivial) LCD MRD codes of length n. Examples are provided for demonstration of results obtained. [ABSTRACT FROM AUTHOR]

We study LCD (linear complementary dual) and ACD (additive complementary dual) codes over a noncommutative non-unital ring E with four elements. This is the first attempt to construct LCD codes over a noncommutative non-unital ring. We show that free LCD codes over E are directly related to binary LCD codes. We introduce ACD codes over E. They include free LCD codes over E as a special case. These facts imply that LCD and ACD codes over E are worth studying. In particular, we characterize a free LCD E-code C in terms of a binary generator matrix G. We also define an ACD code over E, called a left-ACD code. We give several conditions for the existence of left-ACD codes. [ABSTRACT FROM AUTHOR]

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Linear complementary dual (LCD) codes are linear codes satisfying C ∩ C ⊥ = { 0 }. Researchers have proved to construct linear codes over finite fields F q , q>3, is equivalent to construct LCD codes. This means that the investigation of binary and ternary LCD codes is the only remaining open problem. In this work, we propose new constructions of binary and ternary LCD codes by using matrix-product codes and prove that binary and ternary LCD codes from matrix-product codes are asymptotically good. Then we construct some new and good binary and ternary LCD codes using matrix-product matrices. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, q = pm, R = Fq + uFq with u2 = 1, and S = Fq + uFq + vFq + uvFq with u2 = 1, v2 = 1, uv = vu. In this paper, FqRS-cyclic codes over FqRS are studied. As an application, we present a construction of quantum error-correcting codes (QECCs) from the FqRS-cyclic codes over FqRS, which provides new QECCs. We also consider linear complementary dual (LCD) codes from the FqRS-cyclic codes over FqRS. Among others, we construct a Gray map over FqRS and discuss the Gray images of FqRS-cyclic codes over Fq.

Linear complementary dual codes (shortly LCD codes) are codes whose intersections with their dual codes are trivial. These codes were first introduced by Massey in 1992. Nowadays, LCD codes are extensively studied in the literature and widely applied in data storage, cryptography, etc. In this paper, we prove some properties of binary LCD codes using their shortened and punctured codes. We also present some inequalities for the largest minimum weight d LCD (n , k) of binary LCD [n, k] codes for given length n and dimension k. Furthermore, we give two tables with the values of d LCD (n , k) for k ≤ 32 and n ≤ 40 , and two tables with classification results. [ABSTRACT FROM AUTHOR]

A Goppa code over $\Bbb F_{q^{m}}$ is a well-known subclass of algebraic error-correcting code. If $m=1$ , then it is a generalized Reed-Solomon(GRS) code and its dual code is called a GRS code via a Goppa code. In this paper, we give a necessary and sufficient condition that the dual codes of GRS codes via (expurgated) Goppa codes are also GRS codes via Goppa codes. Under the above condition, we show that the hulls of GRS codes via Goppa codes are still GRS codes via Goppa codes. As an application, we characterize LCD GRS codes and self-dual GRS codes under the above condition. Some numerical examples are also presented to illustrate our main results. Moreover, we also apply our result to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs with arbitrary parameters. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with order q and D2n be the dihedral group with 2n elements, and gcd(q, 2n) = 1. In this article, the authors give precise descriptions and enumerations of linear complementary dual (LCD) codes and self-orthogonal codes in the finite dihedral group algebras F q [ D 2 n ] . Some numerical examples are also presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

For an integer 1, m ≥ we study cyclic codes of length ℓ over a commutative non-chain ring F2m + uF2m, where u² = u. With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length ℓ with respect to their generator polynomials and prove that F2m--image of an LCD code of length ℓ is an LCD code of length 2ℓ. Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m. [ABSTRACT FROM AUTHOR]

Bazzi and Mitter showed that binary dihedral group codes are asymptotically good. In this paper we prove that the dihedral group codes over any finite field with strong duality property are asymptotically good. If the characteristic of the field is even, self-dual dihedral group codes are asymptotically good. If the characteristic of the field is odd, maximal self-orthogonal dihedral group codes and LCD dihedral group codes are asymptotically good. [ABSTRACT FROM AUTHOR]

Both linear complementary dual (LCD) codes and maximum distance separable (MDS) codes have good algebraic structures, and they have interesting practical applications such as communication systems, data storage, quantum codes, and so on. So far, most of LCD MDS codes have been constructed by employing generalized Reed-Solomon codes. In this paper we construct some classes of new Euclidean LCD MDS codes and Hermitian LCD MDS codes which are not monomially equivalent to Reed-Solomon codes, called LCD MDS codes of non-Reed-Solomon type. Our method is based on the constructions of Beelen et al. (2017) and Roth and Lempel (1989). To the best of our knowledge, this is the first paper on the construction of LCD MDS codes of non-Reed-Solomon type; any LCD MDS code of non-Reed-Solomon type constructed by our method is not monomially equivalent to any LCD code constructed by the method of Carlet et al. (2018). [ABSTRACT FROM AUTHOR]

𝒯 -Direct codes are an extension to the class of linear codes having complementary duals (LCD codes). Defined over a finite field 𝔽 q , it is comprised of 𝒯 linear codes 𝒞 i ⊆ 𝔽 q n , i = 1 , 2 , ... , 𝒯 with 𝒞 i ∩ 𝒞 i ⊥ = { 0 } , where 𝒞 i ⊥ is the dual with respect to Λ = 𝒞 1 ⊕ 𝒞 2 ⊕ ⋯ ⊕ 𝒞 𝒯 ⊆ 𝔽 q n . A 2-Direct code (𝒞 1 , 𝒞 2) with respect to 𝔽 q n is comprised of only LCD codes: 𝒞 1 ∩ 𝒞 2 = { 0 }. On the contrary, two LCD codes do not set up a 2 -Direct code in general. This paper presents some important and generalized results on 𝒯 -Direct codes, in that it attempts to construct 𝒯 -Direct codes from LCD codes. The class of q -cyclic maximum rank distance (MRD) codes as having complementary duals over 𝔽 q are generalized. Dual bases such as self-dual basis and self-dual normal basis play a crucial role in constructions. Further, construction implausibility of 𝒯 -Direct codes from almost self-dual bases is also dealt. Results obtained are demonstrated through examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the relationship between vectorial (Boolean) functions and cyclic codes in the context of algebraic attacks. We first derive a direct link between the annihilators of a vectorial function (in univariate form) and certain 2 n -ary cyclic codes (which we show that they are LCD codes). We also present some properties of those cyclic codes as well as their weight enumerator. In addition, we generalize the so-called algebraic complement and study its properties. [ABSTRACT FROM AUTHOR]

Given a commutative ring R with identity, a matrix A ∈ M s × l (R) , and linear codes C 1 , ⋯ , C s over R of the same length, this article considers the hull of the matrix-product code [ C 1 ⋯ C s ] A . Consequently, it introduces various sufficient conditions (as well as some necessary conditions in certain cases) under which [ C 1 ⋯ C s ] A is a complementary dual (LCD) code. As an application, LCD matrix-product codes arising from torsion codes over finite chain rings are considered. Moreover, we show the existence of asymptotically good sequences of LCD matrix-product codes over such rings. [ABSTRACT FROM AUTHOR]

Linear complementary dual (LCD) codes provide an optimum linear coding solution for the two-user binary adder channel. LCD codes also can be used against side-channel attacks and fault non-invasive attacks. In this paper, we obtain a lower bound on the distance of binary LCD codes through expanded codes. We give necessary and sufficient conditions to extend binary [ n , k ] LCD codes to binary [ n + 1 , k ] and [ n + 1 , k + 1 ] LCD codes. A sufficient condition for a binary [ n , k , d − 1 ] LCD code constructed from a binary [ n + 1 , k , d ] LC D o , e code is also given. Finally, we construct some new binary LCD codes by using our LCD bounds and some methods such as puncturing, shortening, expanding and extension. In particular, we improve some previously known range of d LCD (n , k) of lengths 38 ≤ n ≤ 40 and dimensions 9 ≤ k ≤ 15. We also obtain some values or range of d LCD (n , k) with 41 ≤ n ≤ 50 and 6 ≤ k ≤ n − 6. [ABSTRACT FROM AUTHOR]

Locally recoverable (LRC) codes and linear complementary dual (LCD) codes are both useful in distributed storage systems. In this letter, we establish a connection between LRC codes and LCD codes. We derive some conditions on the construction of cyclic codes with (r,δ) locality and codes with hierarchical locality so that they are also LCD codes. Some constructed codes are optimal. These codes may be useful in simultaneously providing local recoverability and security against certain attacks in distributed storage systems. [ABSTRACT FROM AUTHOR]

Linear complementary dual (LCD) codes have attracted much attention in recent years due to their applications in implementations against side-channel attacks and fault injection attacks. Comparing coset leaders, we introduce the new concept of absolute coset leaders, which provides advantages for constructing LCD BCH codes. We then give explicit presentations for the largest, second largest and third largest absolute coset leaders and use them to construct binary LCD BCH codes. Lastly, we determine the parameters of these LCD BCH codes. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements, D 2 n , r a generalized dihedral group with gcd (2 n , q) = 1 , and F q [ D 2 n , r ] a generalized dihedral group algebra. Firstly, an explicit expression for primitive idempotents of F q [ D 2 n , r ] is determined, which extends the results of Brochero Martínez (Finite Fields Appl 35:204–214, 2015). Secondly, all linear complementary dual (LCD) codes and self-orthogonal codes in F q [ D 2 n , r ] are precisely described and counted. Some numerical examples are also presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

Linear codes with complementary duals intersect with their duals trivially. Multinegacirculant codes that are complementary dual are characterized algebraically and some good codes are found in this family. Exact enumeration is performed for indices 2 and 3, whereas special choices of the co-index and base field size are needed for higher indices. Asymptotic existence results are derived for the special class of such codes that have co-index a power of two by means of Dickson polynomials. This shows that there are infinite families of complementary dual multinegacirculant codes with relative distance satisfying a modified Gilbert-Varshamov bound. [ABSTRACT FROM AUTHOR]

Double circulant codes of length 2n over the non-local ring R = F q + u F q , u 2 = u , are studied when q is an odd prime power, and − 1 is a square in F q . Double negacirculant codes of length 2n are studied over R when n is even, and q is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length 2n is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length 4n over F q are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below for n →∞. The parameters of examples of modest lengths are computed. Several such codes are optimal. [ABSTRACT FROM AUTHOR]