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The classical way of extending an [ n , k , d ] linear code C is to add an overall parity-check coordinate to each codeword of the linear code C. This extended code, denoted by C ‾ (− 1) and called the standardly extended code of C , is a linear code with parameters [ n + 1 , k , d ¯ ] , where d ¯ = d or d ¯ = d + 1. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper. [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]

A pair (n , q) with q a prime power and n a positive integer for which gcd ⁡ (n , q) = 1 is non-standard if one of the following properties holds, which we will show to be equivalent. − The group U n , q of n -th roots-of-unity in the splitting field F q m of x n − 1 over F q is fixed set-wise by an F q -linear map on F q m that is not of the form x ⟶ α x q j . − There is a non-cyclic linear recurring sequence s of period n (so with s not of the form s i = α ξ i for i ≥ 0) with associated characteristic polynomial being irreducible over F q , such that U n , q = { s 0 , ... , s n − 1 }. In that case, the group U n , q is called non-standard by Brison and Nogueira, who studied this phenomenon in a sequence of papers. − A q -ary irreducible cyclic code of length n has "extra" permutation automorphisms (some well-known examples are the (duals of the) Golay codes and binary simplex codes for n ≥ 7). We will refer to such codes as non-standard irreducible cyclic codes or NSIC-codes. We first investigate non-standard linear recurring sequence subgroups and establish the equivalence between the first and second of the above properties. Then we investigate permutation automorphisms of general (repeated-root) cyclic codes and irreducible cyclic codes and establish the equivalence between the first and third of these properties. We also introduce a notion of non-standardness for general cyclic codes, and relate it to our earlier definition of NSIC-codes. This paper is the first part of an expanded version of the arXiv paper [28]. [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. Inspired by the recent work on binary cyclic codes published in Tang and Ding (2022) [26] and the works in Liu et al. (2023) [15] and Liu et al. (2023) [16] , the objectives of this paper are the construction and analysis of four infinite families of ternary cyclic codes with length n = 3 m − 1 for odd m and dimension k ∈ { n / 2 , (n + 2) / 2 } whose minimum distances have a square-root-like lower bound. Their duals have parameters [ n , k ⊥ , d ⊥ ] , where k ⊥ ∈ { n / 2 , (n − 2) / 2 } and d ⊥ also has a square-root-like lower bound. These families of codes and their duals contain distance-optimal cyclic codes. [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. Inspired by the recent work on binary cyclic codes published in Tang and Ding (2022) [14] , the objectives of this paper are the construction and analyses of five infinite families of binary cyclic codes with parameters [ n , k ] and (n − 6) / 3 ≤ k ≤ 2 (n + 6) / 3. Three of the five families of binary cyclic codes and their duals have a very good lower bound on their minimum distances and contain distance-optimal codes. The other two families of binary cyclic codes are composed of binary duadic codes with a square-root-like lower bound on their minimum distances. As a by-product, two infinite families of self-dual binary codes with a square-root-like lower bound on their minimum distances are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we give an improved bound for mixed exponential sums associated to good cyclic codes. [ABSTRACT FROM AUTHOR]

Abstract We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F 2 + u F 2 , using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper. Highlights • A novel bordered construction for self-dual codes can be obtained using groups rings. • Extremal binary self-dual codes of different lengths can be constructed using the constructions. • 41 new additions to the set of known extremal binary self-dual codes of length 68 have been made. • There is a strong connection between groups rings and constructions for self-dual codes. [ABSTRACT FROM AUTHOR]

Abstract This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If RG is a group ring, where R is commutative and S is a set of generators of G then necessary and sufficient conditions on a map from S to RG are established, such that the map can be extended to an R -derivation of RG. Derivations are shown to be trivial for semisimple group algebras of abelian groups. The derivations of finite group algebras are constructed and listed in the commutative case and in the case of dihedral groups. In the dihedral case, the inner derivations are also classified. Lastly, these results are applied to construct well known binary codes as images of derivations of group algebras. [ABSTRACT FROM AUTHOR]

Abstract Under research for nearly sixty years, Bose–Chaudhuri–Hocquenghem (BCH) codes have played increasingly important roles in many applications such as communication, data storage and information security. However, the dimension and minimum distance of BCH codes have been seldom solved by now because of their intractable characteristics. The objective of this paper is to study the dimensions of some binary BCH codes with length n = 2 m + 1. Many new techniques are employed to investigate the coset leaders modulo n. For m = 2 t + 1 , 4 t + 2 , 8 t + 4 and m ≥ 10 , the first five largest coset leaders modulo n are determined, and the dimensions of some BCH codes of length n with designed distance δ > 2 ⌈ m 2 ⌉ are presented. These new skills and results may be helpful to study other types of cyclic codes over finite fields. [ABSTRACT FROM AUTHOR]

Abstract In this paper we give new constraints on the distance distribution of doubly constant-weight (binary) codes. These constraints improve the linear programming bound on sizes of doubly constant-weight codes. Computations are done for all codes of length n ≤ 28 and all improved upper bounds are shown. We moreover show that the improved upper bounds give rise to further new upper bounds. [ABSTRACT FROM AUTHOR]

Maximum-distance-separable (MDS) codes and self-dual codes are important families of classical codes in coding theory. It is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field F q has been completely solved for q is even. In this paper, for finite field with odd characteristic, we obtain some new results on the existence of MDS self-dual codes by generalized Reed-Solomon (GRS) codes and extended GRS codes. [ABSTRACT FROM AUTHOR]

Binary odd-like duadic codes have parameters [ n , (n + 1) / 2 ] , where n is odd. It is well known that the minimum odd weight of odd-like duadic codes has the lower bound n. The binary quadratic-residue codes and the punctured binary Reed-Muller codes of certain order are two families of binary cyclic codes with parameters [ n , (n + 1) / 2 , d ≥ n ]. It is very hard to construct an infinite family of binary cyclic codes with length n and dimension near (n + 1) / 2 whose minimum distances have a square-root bound. Recently, Tang and Ding constructed an infinite family of binary cyclic codes with parameters [ 2 m − 1 , 2 m − 1 , d ] whose minimum distances have lower bounds close to the square-root bound. The objective of this paper is to construct several infinite families of binary cyclic codes of length 2 m − 1 and dimension near 2 m − 1 whose minimum distance d much exceeds the square-root bound. For m ≥ 9 being odd, an infinite family of binary cyclic codes with parameters [ 2 m , 2 m − 1 , d ≥ 3 ⋅ 2 m − 1 2 − 1 ] is presented. For m ≥ 8 being even, an infinite family of binary cyclic codes with parameters [ 2 m , 2 m − 1 − 1 + m 2 , d ≥ 3 ⋅ 2 m − 2 2 + 1 ] is presented. [ABSTRACT FROM AUTHOR]

Linear codes and t -designs are interactive with each other. It is well known that some t -designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended codes of linear codes holding an infinite family of t -designs with t ≥ 3 are reported in the literature. In this paper, we study the extended codes of the augmented codes of a class of binary cyclic codes with three zeros and their dual codes, and show that those codes hold 3-designs. Furthermore, we obtain some shortened codes from the studied cyclic codes and explicitly determine their parameters. Some of those shortened codes are optimal or almost optimal. [ABSTRACT FROM AUTHOR]

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over F q m are GRS codes (resp., extended GRS codes) when m = 1. In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs. [ABSTRACT FROM AUTHOR]

This paper studies and classifies optimal binary self-dual codes having an automorphism of order 7 with 9 cycles. This classification is done by applying a method for constructing binary self-dual codes with an automorphism of odd prime order p . There are exactly 69781 inequivalent binary self-dual [ 64 , 32 , 12 ] codes with an automorphism of type 7 − ( 9 , 1 ) . As for binary [ 66 , 33 , 12 ] self-dual codes with an automorphism of type 7 − ( 9 , 3 ) there are 1652432 such codes. We also construct more than 4 million new optimal codes of length 68 among which are the first known examples of the very elusive s -extremal self-dual codes. We prove the nonexistence of [ 70 , 35 , 14 ] codes with an automorphism of type 7 − ( 9 , 7 ) . Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before. [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]

Binary constant weight codes have important applications in various topics and have been studied for many years. Optimal or near-optimal binary constant weight codes of small lengths have been determined. In this paper we propose an improvement of the Ericson–Zinoviev construction of binary constant weight codes from q -ary codes. By applying this improvement to Reed–Solomon codes, some new or optimal binary constant weight codes are presented. In particular new binary constant weight codes A ( 64 , 10 , 8 ) ≥ 4112 and A ( 64 , 12 , 8 ) ≥ 522 are constructed. We also give explicitly constructed binary constant weight codes which improve the Gilbert and the Graham–Sloane lower bounds asymptotically in a small range of parameters. Some new binary constant weight codes constructed from algebraic-geometric codes by applying our this improvement are also presented. [ABSTRACT FROM AUTHOR]

The paper begins by giving a counter example to show that the algorithm for construction of new perfect poset codes from a given perfect poset code by removal of a coordinate as given by Lee (2004) [11] does not hold. The algorithm has been improved and generalized to obtain new perfect poset block codes from a given perfect poset block code. The modified necessary and sufficient conditions for the construction of new perfect poset codes have been derived as a particular case. A bound has been obtained on the height of poset P s that turns a given π -code into r -perfect ( P s , π ) -code. We show that there does not exist a poset which admits the binary Simplex code of order 3 to be a 2-perfect poset code. Also, all the poset structures which admit the extended ternary Golay code to be a 3-perfect poset code have been classified. [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]

Recently, linear codes with few weights have been studied extensively. These linear codes have wide applications in secret sharing schemes and authentication codes. In this paper, we introduce a new construction of defining sets using cyclotomic mappings and construct three new classes of binary linear codes with two or three weights. We also explicitly determine the weight distributions of these codes. [ABSTRACT FROM AUTHOR]

An additive cyclic code of length n over F 4 can be defined equivalently as an F 2 [ x ] / 〈 x n + 1 〉 -submodule of F 4 [ x ] / 〈 x n + 1 〉. In this paper we study additive cyclic and complementary dual codes of odd length over F 4 with respect to the trace Hermitian inner product and the trace Euclidean inner product. We characterize subfield subcodes and trace codes of these codes by their generators as binary cyclic codes. [ABSTRACT FROM AUTHOR]

Let R = Z 4 be the integer ring mod 4. A double cyclic code of length ( r , s ) over R is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as R [ x ] -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . In this paper, we determine the generator polynomials of this family of codes as R [ x ] -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . Further, we also give the minimal generating sets of this family of codes as R -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual. [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 give a method to lift binary self-dual codes to the ring R k . The lifting method requires solving a system of linear equations over R k . This technique is applied to [ 14 , 7 , 4 ] binary self-dual code to obtain self-dual codes over R 2 . As Gray images of these codes, a substantial number of [ 56 , 28 , 10 ] self-dual codes are generated. By using the extension theorem given by Bouyuklieva and Bouyukliev in [2] , ten new extremal binary self-dual codes of length 58 with new enumerators are found which were not previously known to exist. [ABSTRACT FROM AUTHOR]

For two positive integers m and k , let C e be a class of cyclic code of length 2 m − 1 over F 2 with three nonzeros γ − 1 , γ − ( 2 k + 1 ) and γ − ( 2 e k + 1 ) for e = 2 or 3, where γ is a primitive element of F 2 m . When m gcd ⁡ ( m , k ) is odd, Kasami in 1971 determined the weight distributions of cyclic codes C 2 and C 3 , which is the same as that of the dual of the triple error-correcting BCH code. This paper obtains the weight distributions of C 2 and C 3 for the case of m gcd ⁡ ( m , k ) being even. [ABSTRACT FROM AUTHOR]

Abstract: Binary optimal codes often contain optimal or near-optimal subcodes. In this paper we show that this is true for the family of self-dual codes. One approach is to compute the optimum distance profiles (ODPs) of linear codes, which was introduced by Luo et al. (2010). One of our main results is the development of general algorithms, called the Chain Algorithms, for finding ODPs of linear codes. Then we determine the ODPs for the Type II codes of lengths up to 24 and the extremal Type II codes of length 32, give a partial result of the ODP of the extended quadratic residue code of length 48. We also show that there does not exist a subcode of for , and we find a first example of a doubly-even self-complementary code. [Copyright &y& Elsevier]

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers. [ABSTRACT FROM AUTHOR]

In this paper, for some fixed q , we construct some new classes of maximum distance separable (MDS) self-dual codes over F q by using (extended) generalized Reed-Solomon codes. Our constructions utilize the additive and/or multiplicative structures of the finite field F q. It reveals that by choosing suitable subsets of F q , more MDS self-dual codes with new parameters can be found. [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 butterfly structure [6,29] and the works of Li, Yue and Fu [21] and Jian, Lin and Feng [19] , we introduce a new defining set with the form of the closed butterfly structure and consequently we obtain three classes of 3-weight binary linear codes and a class of 4-weight binary linear codes whose dual is optimal. The lengths and weight distributions of these four classes of linear codes are completely determined by some detailed calculations on certain exponential sums. Computer experiments show that many (almost) optimal codes can be obtained from our construction. [ABSTRACT FROM AUTHOR]

Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of a linear code has many applications such as on the type II wire-tap channel, dealing with t -resilient functions and trellis or branch complexity of linear codes and so on. In this paper, we present a formula for computing the weight hierarchies of linear codes constructed by the generalized method of defining sets. Then, we construct two classes of binary linear codes with a few weights and determine their weight distributions and weight hierarchies completely. Some codes of them can be used in secret sharing schemes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a mechanism on how to construct long MDS self-dual codes from short ones. These codes are special types of generalized Reed-Solomon (GRS) codes or extended generalized Reed-Solomon codes. The main tool is utilizing additive structure or multiplicative structure on finite fields. By applying this method, more MDS self-dual codes can be constructed. [ABSTRACT FROM AUTHOR]

Group codes are right or left ideals in a group algebra of a finite group over a finite field. Following the ideas of a paper on binary group codes by Bazzi and Mitter in 2006, we prove that group codes over finite fields of any characteristic are asymptotically good. [ABSTRACT FROM AUTHOR]

A recovering set for a coordinate position i in a code is a set R i of other coordinate positions such that the value at the i th position can be recovered by accessing the values at coordinate positions in R i. A locally recoverable (LRC) code with multiple recovering sets is a code in which for every coordinate position there are more than one recovering set. Such codes have been generally studied with the assumption that the recovering sets are pairwise disjoint. Recently Kruglik et al. [1] have presented a construction of LRC codes in which the recovering sets of a coordinate need not be disjoint. In this paper, we present a construction of such type of codes by using the construction of RS-like LRC codes. Further, using a bound given in [1] , we have obtained a bound on the rate of the codes from the present construction. Also, we have presented a sufficient condition for a cyclic code over a finite field to be an LRC code with intersecting recovering sets. [ABSTRACT FROM AUTHOR]

In this paper, three classes of binary linear codes with few weights are proposed from vectorial Boolean power functions, and their weight distributions are completely determined by solving certain equations over finite fields. In particular, a class of simplex codes and a class of first-order Reed-Muller codes can be obtained from our construction by taking the identity map, whose dual codes are Hamming codes and extended Hamming codes, respectively. [ABSTRACT FROM AUTHOR]

We study a construction method of binary reversible self-dual codes in this paper. Reversible codes have good properties in applications, and it is interesting to note that a class of reversible codes is closely connected to BCH codes and LCD codes. We first characterize binary reversible self-dual codes. Using these characteristics of reversible self-dual codes, we find an explicit method for constructing all the binary reversible self-dual codes up to equivalence. Furthermore, using this construction, we obtain nine optimal reversible self-dual codes of length 70 which are all inequivalent, and these codes are all new with respect to binary self-dual codes; they all have the same parameter [ 70 , 35 , 12 ] and their automorphism groups have the same order two. [ABSTRACT FROM AUTHOR]

Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual codes by combining many of the previously used techniques. The purpose of this is to construct self-dual codes that were missed using classical construction techniques by constructing self-dual codes with different automorphism groups. We apply the technique to codes over finite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct some extremal self-dual codes of length 64 and 68, constructing 30 new extremal self-dual codes of length 68. [ABSTRACT FROM AUTHOR]

In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in finding an algebraic structure of conjucyclic codes of length n over the finite field F 4. We show that conjucyclic codes of length n over F 4 are related to binary cyclic codes of length 2 n and show that there is a canonical bijective correspondence between the two sets. We illustrate how the factorization of the polynomial x 2 n + 1 plays a critical role in each setting. Moreover, we construct the generator and parity check matrices of conjucyclic codes of length n over F 4. [ABSTRACT FROM AUTHOR]

In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over F q with designed distance δ = a q m − 1 − 1 (resp. δ = a q m − 1 q − 1) for all 1 ≤ a ≤ q − 1 , where q is a prime power and m > 1 is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results of Yue and Hu [16] , and we give the dimension and, in some cases, the Bose distance for a large designed distance in the range a q m − 1 q − 1 , a q m − 1 q − 1 + T for 0 ≤ a ≤ q − 2 , where T = q m + 1 2 − 1 if m is odd, and T = 2 q m 2 − 1 if m is even. [ABSTRACT FROM AUTHOR]

It is known from the CSS code construction that an [ [ m , 2 k − m , ≥ d ] ] q stabilizer code can be obtained from a (Euclidean) dual-containing [ m , k , d ] q code. In [5] , Blackmore and Norton introduced an interesting code called matrix-product code, which is very useful in constructing new quantum codes of large lengths. Recently, Galindo et al. [16] constructed several classes of stabilizer codes from the dual-containing matrix-product codes of (generalized) Reed-Muller, hyperbolic and affine variety ones. In this paper, we first provide a more general approach to construct dual-containing matrix-product codes and then further study it in two cases. The first case generalizes the result by Galindo et al. and constructs dual-containing matrix-product codes more explicitly since the matrices involved are not restricted to be orthogonal. The second case presents a different way to construct dual-containing matrix-product codes in which some of the constituent codes are not required to be dual-containing. Through the construction of dual-containing matrix-product codes of Reed-Muller and affine variety ones, the CSS code construction and Steane's enlargement, we supply several classes of new stabilizer codes over the fields F 5 , F 7 and F 9 either having minimum distances larger than the ones achieved from the first case or the technique in [16] , or having lengths that are not studied in [16]. [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]

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]