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Abstract This paper presents design of T - Direct codes over GF (2 N) with increased users. The class of T - Direct codes are an extension to the class of linear codes with complementary duals (LCD codes) and can be effectively used over a multiple access channel environment. In this paper, a construction procedure that constructs an n 3 - Direct code from an n - Direct code is described. An n n - Direct codes are also constructed. Further, a recursive construction procedure is presented that constructs n 3 m - Direct codes for m ≥ 0. Finally, T T - Direct codes with constituent codes having variable code rates are obtained. The advantage of these constructions is that it increases the number of constituent codes (users) of an existing T - Direct code, thereby supporting more users in a multi-user environment. [ABSTRACT FROM AUTHOR]

Abstract As a special class of linear codes, minimal linear codes have important applications in secret sharing and secure two-party computation. Constructing minimal linear codes with new and desirable parameters has been an interesting research topic in coding theory and cryptography. Ashikhmin and Barg showed that w min / w max > (q − 1) / q is a sufficient condition for a linear code over the finite field GF (q) to be minimal, where q is a prime power, w min and w max denote the minimum and maximum nonzero weights in the code, respectively. The first objective of this paper is to present a sufficient and necessary condition for linear codes over finite fields to be minimal. The second objective of this paper is to construct an infinite family of ternary minimal linear codes satisfying w min / w max ≤ 2 / 3. To the best of our knowledge, this is the first infinite family of nonbinary minimal linear codes violating Ashikhmin and Barg's condition. [ABSTRACT FROM AUTHOR]

The punctured binary Reed–Muller code is cyclic and was generalised into the punctured generalised Reed–Muller code over GF ( q ) in the literature. The first objective of this paper is to present another generalisation of the punctured binary Reed–Muller code and the binary Reed–Muller code, and analyse these codes. The second objective of this paper is to consider two applications of the new codes in constructing LCD codes and 2-designs. The major motivation of constructing and studying the new codes and their extended codes is the construction of 2-designs, which is an interesting topic in combinatorics. It is remarkable that the family of newly generalised cyclic codes contains a subclass of optimal ternary codes with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] for all m ≥ 2 . Their extended codes have parameters [ 3 m , 3 m − 1 − 2 m , 5 ] for all m ≥ 2 , and are also optimal. [ABSTRACT FROM AUTHOR]

As a generalization of cyclic codes, constacyclic codes is an important and interesting class of codes due to their nice algebraic structures and various applications in engineering. This paper is devoted to the study of the q -polynomial approach to constacyclic codes. Fundamental theory of this approach will be developed, and will be employed to construct some families of optimal and almost optimal codes in this paper. [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth [5]. Let C (1 , e) be a cyclic code of length 3 m − 1 over GF (3) with two nonzeros α and α e , where m and e are given integers, and α is a generator of GF (3 m) ⁎. It is shown that C (1 , e) is optimal with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] if one of the following conditions is met. 1) m ≡ 0 (mod 4) , m ≥ 4 , and e = 3 m 2 + 5. 2) m ≡ 2 (mod 4) , m ≥ 6 , and e = 3 m + 2 2 + 5. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of linear codes over finite fields, viz. skew multi-twisted codes (or skew generalized quasi-twisted codes), which is a generalization of some well-known classes of linear codes such as cyclic codes and quasi-cyclic codes. We also study algebraic structures of skew multi-twisted codes and their Galois duals (i.e., orthogonal complements with respect to the Galois inner product). We also view skew multi-twisted codes as direct sums of certain concatenated codes, which gives rise to a method to construct these codes. We also determine a lower bound on their minimum Hamming distances using their multilevel concatenated structure. Apart from this, we determine the parity-check polynomial of each skew multi-twisted code, and obtain BCH type bounds on their minimum Hamming distances. We also determine generating sets of Galois duals of some skew multi-twisted codes from generating sets of these codes. We also derive necessary and sufficient conditions under which a skew multi-twisted code is (i) Galois self-dual, (ii) Galois self-orthogonal and (iii) Galois LCD (linear with complementary dual). As an application of this result, we also provide enumeration formulae for all Hermitian self-dual, Hermitian self-orthogonal, Hermitian LCD and Euclidean LCD multi-twisted codes over finite fields. We also obtain many linear codes with best known and optimal parameters from 1-generator skew multi-twisted codes over finite fields F 8 and F 9. [ABSTRACT FROM AUTHOR]

Abstract A linear code C with the property that C ∩ C ⊥ = { 0 } is said to be a linear complementary dual, or LCD, code. In this paper, we consider generalized affine Cartesian codes which are LCD. Generalized affine Cartesian codes arise naturally as the duals of affine Cartesian codes in the same way that generalized Reed–Solomon codes arise as duals of Reed–Solomon codes. Generalized affine Cartesian codes are evaluation codes constructed by evaluating multivariate polynomials of bounded degree at points in an m -dimensional Cartesian set over a finite field K and scaling the coordinates. The LCD property depends on the scalars used. Because Reed–Solomon codes are a special case, we obtain a characterization of those generalized Reed–Solomon codes which are LCD along with the more general result for generalized affine Cartesian codes. These results are independent of the characteristic of the underlying field. [ABSTRACT FROM AUTHOR]

Abstract Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of two-weight and three-weight linear codes are presented and their weight distributions are determined using Weil sums. Some of the linear codes obtained are optimal or almost optimal with respect to the Griesmer bound. [ABSTRACT FROM AUTHOR]

Abstract Entanglement-assisted quantum error-correcting (EAQEC, for short) codes use pre-existing entanglements between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQEC code from any classical linear code, unlike standard quantum error-correcting codes, they can only be constructed from classical linear codes which contain their Hermitian dual codes. However, how to determine the parameters of ebits c in EAQEC codes is not an easy task. In this paper, let p be prime and e , k be integers, we construct six classes of EAQEC codes based on k -Galois dual codes over finite fields F p e , where 0 ≤ k < e. The parameter of ebits c of these EAQEC codes can be easily generated algebraically. Furthermore, the six classes of EAQEC codes are of maximal entanglement, most of which have better parameters than current EAQEC codes available. [ABSTRACT FROM AUTHOR]

Abstract Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. The objective of this paper is to present a family of optimal ternary cyclic codes from the Niho-type exponent. Specifically, for an odd integer m and a positive integer r with 4 r ≡ 1 (mod m) , a family of cyclic codes C (u , v) of length 3 m − 1 over GF (3) with two nonzeros β u and β v is studied, where β is a generator of GF (3 m) ⁎ , u = (3 m + 1) / 2 and v = 3 r + 2 is the ternary Niho-type exponent. The parameters of this family of cyclic codes are determined. It turns out that C (u , v) is optimal with respect to the Sphere Packing bound if 9 ∤ m and otherwise almost optimal. Thanks to a recent proof of the Dobbertin–Helleseth–Kumar–Martin conjecture by Katz and Langevin, the dual of this family of cyclic codes is shown to have at most five nonzero weights. [ABSTRACT FROM AUTHOR]

In this paper we present several values for the next-to-minimal weights of projective Reed–Muller codes. We work over F q with q ≥ 3 since in [3] we have determined the complete values for the next-to-minimal weights of binary projective Reed–Muller codes. As in [3] here we also find examples of codewords with next-to-minimal weight whose set of zeros is not in a hyperplane arrangement. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. The main purpose of this paper is to establish and exploit the algebraic structure of some repeated-root two dimensional constacyclic codes of length 2 p s . 2 k over the finite field F p m . Moreover, we study all self-dual repeated-root two-dimensional ( λ , 1 ) -constacyclic codes of length 2 p s . 2 or 2 p s . 2 2 over F p m for λ ∈ { − 1 , 1 } . [ABSTRACT FROM AUTHOR]

In this paper 1 , we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis. [ABSTRACT FROM AUTHOR]

The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the Hartmann–Tzeng bound that works for a wide class of skew cyclic codes. We also provide a practical method for constructing them with designed distance. For skew BCH codes, which are covered by our constructions, we discuss decoding algorithms. Detailed examples illustrate both the theory as the constructive methods it supports. [ABSTRACT FROM AUTHOR]

In this paper, a construction of maximum rank distance (MRD) codes as a generalization of generalized Gabidulin codes is given. The family of the resulting codes is not covered properly by additive generalized twisted Gabidulin codes, and does not cover all twisted Gabidulin codes. When the basis field has more than two elements, this family includes also non-affine MRD codes, and such codes exist for all parameters. Therefore, these codes are the first non-additive MRD codes for most of the parameters. [ABSTRACT FROM AUTHOR]

Relative generalized Hamming weights (RGHWs) of a linear code with respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the linear codes. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes of two nonzeros with respect to its irreducible cyclic subcodes. We give two formulae for RGHWs of the cyclic codes. As applications of the formulae, explicit examples are computed. Moreover, RGHWs of cyclic codes in the examples are very large, comparing with the generalized Plotkin bound of RGHWs. So it guarantees very high security for the secret sharing scheme based on the dual codes. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the linear complementary dual codes (LCD codes for short) to k -Galois LCD codes, and study them by a uniform method. A necessary and sufficient condition for linear codes to be k -Galois LCD codes is obtained, two classes of k -Galois LCD MDS codes are exhibited. Then, necessary and sufficient conditions for λ -constacyclic codes being k -Galois LCD codes are characterized. Some classes of k -Galois LCD λ -constacyclic MDS codes are constructed. Finally, we study Hermitian LCD λ -constacyclic codes, and present a class of Hermitian LCD λ -constacyclic MDS codes. [ABSTRACT FROM AUTHOR]

The locality of locally repairable codes (LRCs) for a distributed storage system is the number of nodes that participate in the repair of failed nodes, which characterizes the repair cost. In this paper, we first determine the locality of MacDonald codes, then propose three constructions of LRCs with r = 1 , 2 and 3 . Based on these results, for 2 ≤ k ≤ 7 and n ≥ k + 2 , we give an optimal linear [ n , k , d ] code with small locality. The distance optimality of these linear codes can be judged by the codetable of M. Grassl for n < 2 ( 2 k − 1 ) and by the Griesmer bound for n ≥ 2 ( 2 k − 1 ) . Almost all the [ n , k , d ] codes ( 2 ≤ k ≤ 7 ) have locality r ≤ 3 except for the three codes, and most of the [ n , k , d ] code with n < 2 ( 2 k − 1 ) achieves the Cadambe–Mazumdar bound for LRCs. [ABSTRACT FROM AUTHOR]

Recently, linear codes constructed from defining sets have been studied extensively. They may have excellent parameters if the defining set is chosen properly. Let m > 2 be a positive integer. For an odd prime p , let r = p m and Tr be the absolute trace function from F r onto F p . In this paper, we give a construction of linear codes by defining the code C D = { ( Tr ( a x ) ) x ∈ D : a ∈ F r } , where D = { x ∈ F r : Tr ( x ) = 1 , Tr ( x 2 ) = 0 } . Its complete weight enumerator and weight enumerator are determined explicitly by employing cyclotomic numbers and Gauss sums. However, we find that the code is optimal with respect to the Griesmer bound provided that m = 3 . In fact, it is MDS when m = 3 . Moreover, the codes presented have higher rate compared with other codes, which enables them to have essential applications in areas such as association schemes and secret sharing schemes. [ABSTRACT FROM AUTHOR]

Recently some special type of mixed alphabet codes that generalize the standard codes has attracted much attention. Besides Z 2 Z 4 -additive codes, Z 2 Z 2 [ u ] -linear codes are introduced as a new member of such families. In this paper, we are interested in a new family of such mixed alphabet codes, i.e., codes over Z 2 Z 2 [ u 3 ] where Z 2 [ u 3 ] = { 0 , 1 , u , 1 + u , u 2 , 1 + u 2 , u + u 2 , 1 + u + u 2 } is an 8-element ring with u 3 = 0 . We study and determine the algebraic structures of linear and cyclic codes defined over this family. First, we introduce Z 2 Z 2 [ u 3 ] -linear codes and give standard forms of generator and parity-check matrices and later we present generators of both cyclic codes and their duals over Z 2 Z 2 [ u 3 ] . Further, we present some examples of optimal binary codes which are obtained through Gray images of Z 2 Z 2 [ u 3 ] -cyclic codes. [ABSTRACT FROM AUTHOR]

Linear codes with a few weights are very important in coding theory and have attracted a lot of attention. In this paper, we present a construction of q -ary linear codes from trace and norm functions over finite fields. The weight distributions of the linear codes are determined in some cases based on Gauss sums. It is interesting that our construction can produce optimal or almost optimal codes. Furthermore, we show that our codes can be used to construct secret sharing schemes with interesting access structures and strongly regular graphs with new parameters. [ABSTRACT FROM AUTHOR]

Let r i be positive integers and R i = Z 2 [ x ] / 〈 x r i − 1 〉 for 1 ≤ i ≤ ℓ . Denote R = R 1 × R 2 × ⋯ × R ℓ . Generalized quasi-cyclic (GQC) code C of length ( r 1 , r 2 , … , r ℓ ) over Z 2 can be viewed as Z 2 [ x ] -submodule of R . In this paper, we investigate the algebraic structure of C by presenting its normalized generating set. We also present a method to determine the normalized generating set of the dual code of C , which is derived from the normalized generating set of C . [ABSTRACT FROM AUTHOR]

Cyclic Reed–Solomon codes, a type of BCH codes, are widely used in consumer electronics, communication systems, and data storage devices. This fact demonstrates the importance of BCH codes – a family of cyclic codes – in practice. In theory, BCH codes are among the best cyclic codes in terms of their error-correcting capability. A subclass of BCH codes are the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The objective of this paper is to determine the dimension and minimum distances of two classes of narrow-sense primitive BCH codes with designed distances δ = ( q − 1 ) q m − 1 − 1 − q ⌊ ( m − 1 ) / 2 ⌋ and δ = ( q − 1 ) q m − 1 − 1 − q ⌊ ( m + 1 ) / 2 ⌋ . The weight distributions of some of these BCH codes are also reported. As will be seen, the two classes of BCH codes are sometimes optimal and sometimes among the best linear codes known. [ABSTRACT FROM AUTHOR]

The generalized Hamming weights of a linear code have been extensively studied since Wei first use them to characterize the cryptography performance of a linear code over the wire-tap channel of type II. In this paper, we investigate the generalized Hamming weights of three classes of linear codes constructed through defining sets and determine them partly for some cases. Particularly, in the semiprimitive case we solve a problem left in Yang et al. (2015) [30] . [ABSTRACT FROM AUTHOR]

The generalized Hamming weights (GHWs) are fundamental parameters of linear codes. GHWs are of great interest in many applications since they convey detailed information of linear codes. In this paper, we continue the work of [10] to study the GHWs of a family of cyclic codes with arbitrary number of nonzeroes. The weight hierarchy is determined by employing a number-theoretic approach. [ABSTRACT FROM AUTHOR]

Cyclic codes have attracted a lot of research interest for decades as they have efficient encoding and decoding algorithms. In this paper, for an odd prime p , we investigate two classes of p -ary cyclic codes for special cases and determine their weight distributions explicitly. The results show that both codes have at most five nonzero weights. [ABSTRACT FROM AUTHOR]

In this paper, by analyzing the solutions of certain equations over F 3 m , we present four classes of optimal ternary cyclic codes with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] . It is shown that some recent work on this class of optimal ternary cyclic codes are special cases of our results. [ABSTRACT FROM AUTHOR]

Constacyclic codes form a well-known class of linear codes, and are generalizations of cyclic and negacyclic codes. In this paper, we determine generator polynomials of all constacyclic codes of length 4 ℓ m p n over the finite field F q with q elements, where p , ℓ are distinct odd primes, q is a power of p and m , n are positive integers. We also determine their dual codes, and list all self-dual constacyclic codes of length 4 ℓ m p n over F q . [ABSTRACT FROM AUTHOR]

Two-weight linear codes have many wide applications in authentication codes, association schemes, strongly regular graphs, and secret sharing schemes. In this paper, we present two classes of two-weight binary or ternary linear codes. In some cases, they are optimal or almost optimal. They can also be used to construct secret sharing schemes. [ABSTRACT FROM AUTHOR]

In this paper, we study cyclic codes and constacyclic codes with shift constant ( 2 + u ) over R = Z 4 + u Z 4 , where u 2 = 1 . We determine the form of the generators of the cyclic codes over this ring and their spanning sets. Considering their Z 4 images, we prove that the Z 4 -image of a ( 2 + u ) -constacyclic code of odd length is a cyclic code over Z 4 . We also present many examples of cyclic codes over R whose Z 4 -images have better parameters than previously best-known Z 4 -linear codes. [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Let m = 2 ℓ + 1 for an integer ℓ ≥ 1 and π be a generator of GF ( 3 m ) ⁎ . In this paper, a class of cyclic codes C ( u , v ) over GF ( 3 ) with two nonzeros π u and π v is studied, where u = ( 3 m + 1 ) / 2 , and v = 2 ⋅ 3 ℓ + 1 is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over GF ( 3 m ) , the cyclic code C ( u , v ) is shown to have minimal distance four, which is the best minimal distance for any linear code over GF ( 3 ) with length 3 m − 1 and dimension 3 m − 1 − 2 m according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied. [ABSTRACT FROM AUTHOR]

Cyclic codes are an important subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics. In this paper, two families of optimal ternary cyclic codes are presented. The first family of cyclic codes has parameters [3m-1,3m-1-2m,4] and contains a class of conjectured cyclic codes and several new classes of optimal cyclic codes. The second family of cyclic codes has parameters [3m-1,3m-2-2m,5] and contains a number of classes of cyclic codes that are obtained from perfect nonlinear functions over F3m, where m>1 and is a positive integer. [ABSTRACT FROM AUTHOR]

Minimal linear codes have important applications in secret sharing schemes and secure multi-party computation, etc. In this paper, we study the minimality of a kind of linear codes over GF (p) from Maiorana-McFarland functions. We first obtain a new sufficient condition for this kind of linear codes to be minimal without analyzing the weights of its codewords, which is a generalization of some works given by Ding et al. in 2015. Using this condition, it is easy to verify that such minimal linear codes satisfy w min w max ≤ p − 1 p for any prime p , where w min and w max denote the minimum and maximum nonzero weights in a code, respectively. Then, by selecting the subsets of GF (p) s , we present two new infinite families of minimal linear codes with w min w max ≤ p − 1 p for any prime p. In addition, the weight distributions of the presented linear codes are determined in terms of Krawtchouk polynomials or partial spreads. [ABSTRACT FROM AUTHOR]

Linear complementary dual codes were defined by Massey in 1992, and were used to give an optimum linear coding solution for the two user binary adder channel. In this paper, we define the analog of LCD codes over fields in the ambient space with mixed binary and quaternary alphabets. These codes are additive, in the sense that they are additive subgroups, rather than linear as they are not vector spaces over some finite field. We study the structure of these codes and we use the canonical Gray map from this space to the Hamming space to construct binary LCD codes in certain cases. We give examples of such binary LCD codes which are distance-optimal, i.e., they have the largest minimum distance among all binary LCD codes with the same length and dimension. [ABSTRACT FROM AUTHOR]

Linear codes with few weights have applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the works of Heng and Yue (2016) [14] and Tan, Zhou, Tang and Helleseth (2017) [25] , we extend Tan, Zhou, Tang and Helleseth's work to obtain a class of optimal 1-weight binary linear codes, new classes of 2-weight and 3-weight p -ary linear codes and a class of 4-weight binary linear codes. The lengths and weight distributions of the t -weight linear codes, where t = 1 , 2 , 3 , are closed-form expressions of Kloosterman sums over finite prime fields, and are completely determined when p = 2 and p = 3. [ABSTRACT FROM AUTHOR]

The component-wise or Schur product C ⁎ C ′ of two linear error-correcting codes C and C ′ over certain finite field is the linear code spanned by all component-wise products of a codeword in C with a codeword in C ′. When C = C ′ , we call the product the square of C and denote it C ⁎ 2. Motivated by several applications of squares of linear codes in the area of cryptography, in this paper we study squares of so-called matrix-product codes, a general construction that allows to obtain new longer codes from several "constituent" codes. We show that in many cases we can relate the square of a matrix-product code to the squares and products of their constituent codes, which allow us to give bounds or even determine its minimum distance. We consider the well-known (u , u + v) -construction, or Plotkin sum (which is a special case of a matrix-product code) and determine which parameters we can obtain when the constituent codes are certain cyclic codes. In addition, we use the same techniques to study the squares of other matrix-product codes, for example when the defining matrix is Vandermonde (where the minimum distance is in a certain sense maximal with respect to matrix-product codes). [ABSTRACT FROM AUTHOR]

Let F 2 m be a finite field of cardinality 2 m , R = F 2 m + u F 2 m (u 2 = 0) and s , n be positive integers such that n is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring R of length 2 s n and provide a calculation method to obtain all distinct codes. Moreover, we obtain a clear formula to count the number of all these self-dual cyclic codes. As an application, self-dual and 2-quasi-cyclic codes over F 2 m of length 2 s + 1 n can be obtained from self-dual cyclic code over R of length 2 s n and by a Gray map preserving orthogonality and distances from R onto F 2 m 2. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. The objective of this paper is to investigate the parameters of some q -ary cyclic codes of length n = q 2 m − 1 q + 1. The dimensions of these codes are settled and the lower bounds on their minimum distances are presented by employing the Hartmann-Tzeng bound or BCH bound. Moreover, when q = 2 , 3 , we give the largest coset leaders δ 1 modulo n and investigate the parameters of the BCH codes C (q , n , δ 1 , 1). [ABSTRACT FROM AUTHOR]