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In this paper, δ -dual codes over finite commutative semi-simple rings are defined as a generalization of dual codes over finite commutative semi-simple rings. Some properties of δ -dual codes are given. We present necessary and sufficient conditions for a λ -constacyclic code of length n to be δ -self-dual, δ -self-orthogonal, δ -dual-containing, δ -LCD over finite commutative semi-simple rings. We also study the δ -dual of skew Θ - λ -constacyclic codes over finite commutative semi-simple rings. Among others, we also give necessary and sufficient conditions for a skew Θ - λ -constacyclic code of any length n to be δ -self-dual, δ -self-orthogonal, δ -dual-containing, δ -LCD over finite commutative semi-simple rings. [ABSTRACT FROM AUTHOR]

In this paper we construct distance-regular graphs admitting a vertex transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups M 11 , M 12 , M 22 , M 23 and M 24 . From the binary code spanned by an adjacency matrix of the strongly regular graph with parameters (176,70,18,34) we obtain block designs having the full automorphism groups isomorphic to the Higman-Sims finite simple group. Moreover, from that code we obtain eight 2-designs having the full automorphism group isomorphic to M 22 , whose existence cannot be explained neither by the Assmus-Mattson theorem nor by a transitivity argument. Further, we discuss a possibility of permutation decoding of the codes spanned by adjacency matrices of the graphs constructed and find small PD-sets for some of the 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. In this paper, two classes of cyclic codes whose duals have two zeros are presented. The weight distributions of these cyclic codes are settled with the help of Gaussian periods. The duals of one class of cyclic codes are also studied. Some of the cyclic codes presented in this paper are optimal in the sense that they meet some bounds of linear codes. [ABSTRACT FROM AUTHOR]

Linear codes with a few weights have nice applications in communication, secret sharing schemes, authentication codes, association schemes, block designs and so on. Projective binary linear codes are one of the most important subclasses of linear codes for practical applications. The objective of this paper is to construct projective binary linear codes with some special Boolean functions. Four families of binary linear codes with three or four weights are derived and the parameters of their duals are also determined. It turns out that the duals of these codes are optimal or almost optimal with respect to the sphere-packing bound. As applications, the codes presented in this paper can be used to construct association schemes and secret sharing schemes with interesting access structures. [ABSTRACT FROM AUTHOR]

Let θ be an automorphism on a finite field F q. In this paper, we give a way to construct and enumerate Euclidean self-dual θ -cyclic codes of length n over F q when n is even and gcd (n , | θ |) = 1. The restriction gcd (n , | θ |) = 1 implies that the θ -cyclic codes are in fact cyclic codes and q = 2 m , for some integer m ≥ 1. The construction and enumeration are done by analyzing the orbits of cyclotomic cosets under a multiplier map induced by θ. [ABSTRACT FROM AUTHOR]

In this paper we generalize the construction of binary self-orthogonal codes obtained from weakly self-orthogonal designs described in Tonchev (J Combinat Theory Ser A 52:197-205, 1989) in order to obtain self-orthogonal codes over an arbitrary field. We extend construction self-orthogonal codes from orbit matrices of self-orthogonal designs and weakly self-orthogonal 1-designs such that block size is odd and block intersection numbers are even described in Crnković (Adv Math Commun 12:607–628, 2018). Also, we generalize mentioned construction in order to obtain self-orthogonal codes over an arbitrary field. We construct weakly self-orthogonal designs invariant under an action of Mathieu group M 11 and, from them, binary self-orthogonal codes. [ABSTRACT FROM AUTHOR]

Let m = 8 k and α be a primitive element of the finite field G F (2 m) , where k ≥ 2 is an integer. In this paper, a class of binary cyclic codes C (u , v) of length 2 m - 1 with two nonzeros α - u and α - v is studied, where (u , v) = (1 , (2 m - 1) / 17) . It turns out that C (u , v) has parameters [ 2 m - 1 , 2 m - m - 9 , 4 ] and is distance-optimal with respect to the Sphere Packing bound. The weight distribution of the dual of C (u , v) is also completely determined based on some results on Gaussian periods. [ABSTRACT FROM AUTHOR]

In 1990 Cooper suggested to use Groebner bases' computations to decode cyclic codes and his idea gave rise to many research papers. In particular, as proved by Sala-Orsini, once defined the polynomial ring whose variables are the syndromes, the locations and the error values and considered the syndrome ideal, only one polynomial of a lexicographical Groebner basis for such ideal is necessary to decode (the general error locator polynomial, a.k.a. GELP). The decoding procedure only consists in evaluating this polynomial in the syndromes and computing its roots: the roots are indeed the error locations. A possible bottleneck in this procedure may be the evaluation part, since a priori the GELP may be dense. In this paper, focusing on binary cyclic codes with length n = 2 m - 1 , correcting up to two errors, we give a Groebner-free, sparse analog of the GELP, the half error locator polynomial (HELP). In particular, we show that it is not necessary to compute the whole Groebner basis for getting such kind of locator polynomial and we construct the HELP, studying the quotient algebra of the polynomial ring modulo the syndrome ideal by a combinatorial point of view. The HELP turns out to be computable with quadratic complexity and it has linear growth in the length n of the code: O n + 1 2 . [ABSTRACT FROM AUTHOR]

In this paper, we provide new methods and algorithms to construct Euclidean self-dual codes over large finite fields. With the existence of a dual basis, we study dual preserving linear maps, and as an application, we use them to construct self-orthogonal codes over small finite prime fields using the method of concatenation. Many new optimal self-orthogonal and self-dual codes are obtained. [ABSTRACT FROM AUTHOR]

As a special type of linear codes, minimal linear codes have important applications in secret sharing. Up to now, only a few infinite families of minimal binary linear codes with wmin/wmax≤1/2 were reported in the literature, while vast knowledge exists on the ones with wmin/wmax>1/2. Herein, wmin and wmax respectively denote the minimum and maximum nonzero Hamming weights in a linear code. Recently, several classes of linear codes with certain properties were constructed by Zhou et al. from a generic construction. The objective of this paper is to obtain four families of minimal binary linear codes with wmin/wmax≤1/2 from those linear codes proposed by Zhou et al. The parameters of our minimal linear codes are quite different from known ones. Based on the properties of Krawtchouk polynomials, the weight distributions of all these four families of binary linear codes are established. [ABSTRACT FROM AUTHOR]

In this paper, we study cyclic codes and their duals over the local Frobenius non-chain ring R = F 2 [ u , v ] / ⟨ u 2 = v 2 , u v ⟩ , and we obtain optimal binary linear codes with respect to the homogeneous weight over R via a Gray map. Moreover, we characterize DNA codes as images of cyclic codes over R. [ABSTRACT FROM AUTHOR]

In this work we define and study binary codes C q , k and C q , k ¯ obtained from neighborhood designs of Paley-type bipartite graphs P(q, k) and their complements, respectively for q an odd prime. We prove that for some values of q and k the codes C q , k are self-dual and the codes C q , k ¯ are self-orthogonal. Most of these codes tend to be with optimal or near optimal parameters. Next, we extend the codes C q , k to get doubly even self dual codes and find that most of these codes are extremal. [ABSTRACT FROM AUTHOR]

Let G be a nontrivial abelian group and let A 1 , ... , A h ( h ≥ 2 ) be nonempty subsets of G. We say that A 1 , ... , A h is a complete decomposition of G of order h if A 1 + ⋯ + A h = G and A i ∩ A j = ∅ for i , j = 1 , ... , h ( i ≠ j ). In this paper we consider the case G is the cyclic group Z n and determine the values of h for which a complete decomposition of Z n of order h exists. The result is then extended to the case G is a finite abelian group. We also investigate the existence of complete decompositions of Z n where the cardinality of each set in the decomposition is a prescribed integer ≥ 2 . As an application, we describe a way to construct codes over a binary alphabet using a construction of a complete decomposition of cyclic groups. [ABSTRACT FROM AUTHOR]

Let $$n, m \ge 2$$ be integers. The cartesian, categorical and lexicographic products of m copies of the n-cycle denoted by $$C_n$$ all have as their vertex-set $$\{0, 1, \ldots , n-1\}^m$$ , with adjacency defined variously. In this paper the binary codes generated by the row span of adjacency matrices of the cartesian, categorical and lexicographic products of m copies of $$C_n$$ are examined. Full and partial PD-sets were also found for the various codes. [ABSTRACT FROM AUTHOR]

In the past decades, linear codes with a few weights have been extensively studied for their applications in space communication, data storage and cryptography etc. We construct several classes of binary linear codes and determine their weight distributions. Most of these codes can be used in secret sharing schemes. [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]

We define additive skew G-codes over finite fields and discuss several dualities attached to these codes. We examine the properties of self-dual skew G-codes and in particular we show that the dual, under any duality, of an additive skew G-code is also an additive skew G-code. Additionally, we propose a matrix construction for additive skew G-codes and use it to construct several examples of extremal self-dual additive skew G-codes over the finite field F 4 . Such codes have a strong connection to quantum error correcting codes. [ABSTRACT FROM AUTHOR]

Let K = Q (ζ 8) be the complex multiplication field over Q of extension degree 4. We give an integral lattice construction on Q (ζ 8) induced from binary codes. We define a theta series using these lattices and discuss its relation with the complete weight enumerator of a binary code. If C is a binary Type II code of length l, we find that the complete weight enumerator of C gives a Jacobi form of weight l and the index 2l over the maximal totally real subfield k = Q (ζ 8 + ζ 8 - 1) of K. Also, we see that Hilbert-Siegel modular form of weight l and genus g can be seen in terms of the complete joint weight enumerator for codes C j , 1 ≤ j ≤ g over F 2 . [ABSTRACT FROM AUTHOR]

In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring M k (R) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring M k (R) are one sided ideals in the group matrix ring M k (R) G and the corresponding codes over the ring R are G k -codes of length kn. Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes. [ABSTRACT FROM AUTHOR]

The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers x = 1 , y = 3 is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if p < 41 is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then p ≤ 5 . The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism ϕ of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under ϕ . [ABSTRACT FROM AUTHOR]

We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group Z p e . They exist for all lengths when p is prime and e is even; all even lengths when p is an odd prime with p ≡ 1 (mod 4) and e is odd with e > 1 ; and all lengths that are 0 (mod 4) when p is an odd prime with p ≡ 3 (mod 4) and e is odd with e > 1. [ABSTRACT FROM AUTHOR]