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In this study, we determine the structure of cyclic codes over the ring Z q [ u ] / 〈 u 2 〉 which is isomorphic to R = Z q + u Z q where q = p s , p is a prime, s is a positive integer, and u 2 = 0. This is equivalent to determining the algebraic structure of ideals of the polynomial quotient ring R [ x ] / 〈 x n − 1 〉 , which is addressed in this paper completely. By establishing the structure of ideals of R [ x ] / 〈 x n − 1 〉 with gcd ⁡ (p , n) = 1 , we present an exact formula that enumerates the number of ideals of this ring that leads to the enumeration of cyclic codes over this ring. Finally, we consider and explore some special families of cyclic codes for some specific q and determine their size. [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]

In this paper, we study \BBZ2\BBZ4-additive cyclic codes. These codes are identified as \BBZ4[x]-submodules of the ring Rr,s=\BBZ2[x]/\langle x^{r}-1\rangle\times\BBZ4\left[x\right]/\langle x^{s}-1\rangle. The algebraic structure of this family of codes is studied and a set of generator polynomials for this family as a \BBZ4[x]-submodule of the ring Rr,s is determined. We show that the duals of \BBZ2\BBZ4-additive cyclic codes are also cyclic. We also present an infinite family of Maximum Distance separable with respect to the singleton bound codes. Finally, we obtain a number of binary linear codes with optimal parameters from the \BBZ2\BBZ4-additive cyclic codes. [ABSTRACT FROM AUTHOR]

In this paper, we study a 2-dimensional cellular automaton generated by a new local rule with the nearest neighborhoods and prolonged next nearest neighborhoods under periodic boundary condition over the ternary field (Z3). We obtain the rule matrix of this cellular automaton and characterize this family by exploring some of their important characteristics. We get some recurrence equations which simplifies the computation of the rank of the rule matrix related to the 2-dimensional cellular automaton drastically. Next, we propose an algorithm to determine the rank of the rule matrix. Finally, we conclude by presenting an application to error correcting codes. [ABSTRACT FROM AUTHOR]

Abstract: The structure of DNA is used as a model for constructing good error correcting codes and conversely error correcting codes that enjoy similar properties with DNA structure are also used to understand DNA itself. Recently, naturally four element sets are used to model DNA by some families of error correcting codes. Hence the structure of such codes has been studied. In this paper, the authors first relate DNA pairs with a special 16 element ring. Then, the so-called cyclic DNA codes of odd length that enjoy some of the properties of DNA are studied. Their algebraic structure is determined. Further, by introducing a map, a family of cyclic codes over this ring is mapped to DNA codes. Hamming minimum distances are also studied. The paper concludes with some DNA examples obtained via this family of cyclic codes. [Copyright &y& Elsevier]

Abstract: Cellular automata are used to model dynamical phenomena by focusing on their local behavior which depends on the neighboring cells in order to express their global behavior. The geometrical structure of the models suggests the algebraic structure of cellular automata. After modeling the dynamical phenomena, it is sometimes an important problem to be able to move backwards in order to understand it better. This is only possible if cellular automata is reversible. In this paper, 2D finite cellular automata defined by local rules based on hexagonal cell structure are studied. Rule matrix of the hexagonal finite cellular automaton is obtained. The rank of rule matrices representing the 2D hexagonal finite cellular automata via an algorithm is computed. It is a well known fact that determining the reversibility of a 2D cellular automata is a very difficult problem in general. Here, the reversibility problem of this family of 2D hexagonal cellular automata is also resolved completely. [Copyright &y& Elsevier]

Abstract: In this work, the cardinality of the minimal -covers of finite rings with respect to the RT-metric is established. By generalizing the result in Nakaoka and dos Santos (2010) , the minimal cardinalities of 0-short coverings of finite chain rings are calculated. The connection between -short coverings of rings with respect to the RT-metric and the 0-short coverings of rings is demonstrated, and with the help of this connection, the problem of finding the minimal cardinalities of -short coverings of finite chain rings is solved. [Copyright &y& Elsevier]

Abstract: It is a well established fact that -spotty byte error control codes provide a good source for detecting and correcting errors in semiconductor memory systems using high-density RAM chips with wide I/O data (e.g. 8, 16 or 32 bits). Recently, a MacWilliams identity that establishes an important relation between an -spotty weight enumerator of a binary code and its dual has been proven in Kazuyoshi Suzuki et al. (2007) . In this paper, we introduce the -spotty Lee weights and the -spotty Lee weight enumerator of a quaternary code and prove a MacWilliams type identity. [Copyright &y& Elsevier]

Focuses on a study which proved a MacWilliams identity for a ρ complete weight enumerator of linear spaces of matrices with entries from the ring of a finite field with q elements. Definition of the complete weight enumerator; Lemma; Proofs.

Examines the minimum-distance bounds of linear codes. Improvement in minimum distances; Introduction of a generalized version of a Gray map; Definitions of quasi- and nearly quasi-cyclic codes.

Presents a study which examined the ternary quasi-cyclic (QC) codes with better minimum distances. Structure of one-generator QC codes; Codes and generator matrices.

In this work, we study cyclic codes that have generators as Fibonacci polynomials over finite fields. We show that these cyclic codes in most cases produce families of maximum distance separable and optimal codes with interesting properties. We explore these relations and present some examples. Also, we present applications of these codes to secret sharing schemes. [ABSTRACT FROM AUTHOR]

Following the very recent studies on \mathbb Z2\mathbb Z4 -additive codes, \mathbb Z2\mathbb Z2[u] -linear codes have been introduced by Aydogdu et al. In this paper, we introduce and study the algebraic structure of cyclic, constacyclic codes and their duals over the R -module \mathbb {Z}_{2}^\alpha R^\beta where R=\mathbb {Z}_{2}+u\mathbb {Z}_{2}=\left \{{0,1,u,u+1}\right \} is the ring with four elements and u^2=0 . We determine the generating independent sets and the types and sizes of both such codes and their duals. Finally, we present a bound and an optimal family of codes attaining this bound and also give some illustrative examples of binary codes that have good parameters which are obtained from the cyclic codes in \mathbb Z_2^\alpha R^\beta . [ABSTRACT FROM AUTHOR]

In this work we introduce a novel approach to find reversible codes over different alphabets, using so-called coterm polynomials and a module-construction. We obtain many optimal reversible codes with these constructions. In an attempt to apply the constructions to the DNA, we identify k -bases of DNA with elements in the ring R 2 k = F 2 [ u ] / ( u 2 k − 1 ) , and by using a form of coterm polynomials, we are able to solve the reversibility and complement problems in DNA codes over this ring. With a freedom on the choice of k we are able to embed any DNA code in a suitable ring, giving an algebraic structure to the DNA codes. We are also able to find reversible and reversible-complement codes that are not necessarily linear cyclic codes. [ABSTRACT FROM AUTHOR]

Cellular automata have rich computational properties and provide many models in mathematical and physical processes. In this paper, one of the most commonly used neighborhood types of two dimensional (2D) cellular automata which is called the Moore neighborhood in two dimensional integer lattice is considered. We study the characterization of 2D linear cellular automata defined by the Moore neighborhood with periodic and null boundary conditions over ternary fields. Furthermore, we analyze some results of 2D cellular automata defined by the rule number 9840NB and finally also present applications to error correcting codes and image processing field. [ABSTRACT FROM AUTHOR]

In this work, quadratic double and quadratic bordered double circulant constructions are applied to F 4 + u F 4 as well as to F 4 , and as a result, extremal binary self-dual codes of length 56 and 64 are obtained. The binary extension theorems as well as the ring extension version are used to obtain seven extremal self-dual binary codes of length 58, twenty-four extremal self-dual binary codes of length 66 and twenty-nine extremal self-dual binary codes of length 68, all with new weight enumerators, updating the list of all the known extremal self-dual codes in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper quadratic residue codes over the ring are introduced in terms of their idempotent generators. The structure of these codes is studied and it is observed that these codes enjoy similar properties as quadratic residue codes over finite fields. For the case , Euclidean and Hermitian self-dual families of codes as extended quadratic residue codes are considered and two optimal Hermitian self-dual codes are obtained as examples. Moreover, a substantial number of good p-ary codes are obtained as images of quadratic residue codes over in the cases where p is an odd prime. These results are presented in tables. [Copyright &y& Elsevier]

In this work, quadratic residue codes over the ring F 2 + u F 2 + u 2 F 2 with u 3 = u are considered. A duality and distance preserving Gray map from F 2 + u F 2 + u 2 F 2 to F 2 3 is defined. By using quadratic double circulant, quadratic bordered double circulant constructions and their extensions self-dual codes of different lengths are obtained. As Gray images of these codes and their extensions, a substantial number of new extremal self-dual binary codes are found. More precisely, thirty two new extremal binary self-dual codes of length 68, 363 Type I codes of parameters [ 72 , 36 , 12 ] , a Type II [ 72 , 36 , 12 ] code and a Type II [ 96 , 48 , 16 ] code with new weight enumerators are obtained through these constructions. The results are tabulated. [ABSTRACT FROM AUTHOR]

This paper presents a study of two-dimensional hexagonal cellular automata (CA) with periodic boundary. Although the basic construction of a cellular automaton is a discrete model, its global level behavior at large times and on large spatial scales can be a close approximation to a continuous system. Meanwhile CA is a model of dynamical phenomena that focuses on the local behavior which depends on the neighboring cells in order to express their global behavior. The mathematical structure of the model suggests the importance of the algebraic structure of cellular automata. After modeling the dynamical behaviors, it is sometimes an important problem to be able to move backwards on CAs in order to understand the behaviors better. This is only possible if cellular automaton is a reversible one. In the present paper, we study two-dimensional finite CA defined by hexagonal local rule with periodic boundary over the field ℤ3 (i.e. 3-state). We construct the rule matrix corresponding to the hexagonal periodic cellular automata. For some given coefficients and the number of columns of hexagonal information matrix, we prove that the hexagonal periodic cellular automata are reversible. Moreover, we present general algorithms to determine the reversibility of 2D 3-state cellular automata with periodic boundary. A well known fact is that the determination of the reversibility of a two-dimensional CA is a very difficult problem, in general. In this study, the reversibility problem of two-dimensional hexagonal periodic CA is resolved completely. Since CA are sufficiently simple to allow detailed mathematical analysis, also sufficiently complex to produce chaos in dynamical systems, we believe that our construction will be applied many areas related to these CA using any other transition rules. [ABSTRACT FROM AUTHOR]

In this study, we consider the theoretical structure and classification of two-dimensional (2D) three-state uniform cellular automata (CA) based on their visual behavior. The basic features of a CA comprise their discrete dynamic structure and the fact that they are modeled locally, but their behavior can be similar to a continuous system at large time and spatial scales. The geometrical structures of the patterns produced by iterating CA can be considered according to some basic properties. After applying the rules iteratively, it has been shown that CA are capable of producing complex behaviors. Some types of CA exhibit remarkably regular behavior with finite configurations. With simple initial configurations, the patterns generated may be self-replicating (SR), self-similar (SS), or mixed. In this study, we consider the theory of 2D uniform periodic boundary (PB), adiabatic boundary (AB), and reflexive boundary (RB) CA with a von Neumann neighborhood, and applications of image analysis for generating patterns. We investigate a 2D CA under these boundary restrictions for three-state cases, i.e., the ternary field Z 3 . We also study the applications of SR and SS patterns that correspond to the linear CA rules of 2D uniform CA with different boundary cases over Z 3 . The von Neumann neighborhood CA rule (i.e., Rule 2460) is classified into SR and SS types depending on the non-zero boundary values a , b , c , d of the neighboring cells that influence the cells under consideration. Based on the visual appearance of the patterns, we also show that Rule 2460 is sometimes sensitive, depending on the boundary conditions and chaotic behavior. Finally, we conclude by analyzing some results in detail for CA defined by Rules 2460NB, 2460PB, 2460AB, and 2460RB for non-symmetric figures. [ABSTRACT FROM AUTHOR]

This paper studies the theoretical aspects of two-dimensional cellular automata (CAs), it classifies this family into subfamilies with respect to their visual behavior and presents an application to pseudo random number generation by hybridization of these subfamilies. Even though the basic construction of a cellular automaton is a discrete model, its macroscopic behavior at large evolution times and on large spatial scales can be a close approximation to a continuous system. Beyond some statistical properties, we consider geometrical and visual aspects of patterns generated by CA evolution. The present work focuses on the theory of two-dimensional CA with respect to uniform periodic, adiabatic and reflexive boundary CA (2D PB, AB and RB) conditions. In total, there are 512 linear rules over the binary field ℤ2 for each boundary condition and the effects of these CA are studied on applications of image processing for self-replicating patterns. After establishing the representation matrices of 2D CA, these linear CA rules are classified into groups of nine and eight types according to their boundary conditions and the number of neighboring cells influencing the cells under consideration. All linear rules have been found to be rendering multiple self-replicating copies of a given image depending on these types. Multiple copies of any arbitrary image corresponding to CA find innumerable applications in real life situation, e.g. textile design, DNA genetics research, statistical physics, molecular self-assembly and artificial life, etc. We conclude by presenting a successful application for generating pseudo numbers to be used in cryptography by hybridization of these 2D CA subfamilies. [ABSTRACT FROM AUTHOR]

In this paper, we study a special type of quasi-cyclic (QC) codes called skew QC codes. This set of codes is constructed using a noncommutative ring called the skew polynomial ring F[x;θ]. After a brief description of the skew polynomial ring F[x;θ], it is shown that skew QC codes are left submodules of the ring Rsl = (F[x;θ]/(xs - 1))l. The notions of generator and parity-check polynomials are given. We also introduce the notion of similar polynomials in the ring F[x;θ] and show that parity-check polynomials for skew QC codes are unique up to similarity. Our search results lead to the construction of several new codes with Hamming distances exceeding the Hamming distances of the previously best known linear codes with comparable parameters. [ABSTRACT FROM AUTHOR]