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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 the algebraic structure of-additive codes which are-submodules whereis prime, andandare positive integers.-additive codes naturally generalizeand-additive codes which have been introduced recently. The results obtained in this work generalize a great amount of the studies done on additive codes. Especially, we determine the standard forms of generator and parity-check matrices for this family of codes. This leads to the identification of the type and the cardinalities of these codes. Furthermore, we present some bounds on the minimum distance and examples that attain these bounds. Finally, we present some illustrative examples for some special cases. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the lifted polynomials which generate reversible codes overFq, a finite field withqelement. Lifted polynomials are introduced by the authors Oztas and Siap [Lifted polynomials over F16and their applications to DNA codes, Filomat 27(3) (2013), pp. 459–466] overF16. Lifted polynomials have proven to be very advantageous. They are easy to construct and they can be used to construct codes with specific properties such as dimension and the length of codes. We also generalize the 4k-lifted polynomials which lead to reversible and reversible complement DNA codes over. Further we construct examples of codes overF8,F9,F16andF256that have the best possible parameters or attain the Griesmer bound, hence they are optimal codes generated by lifted polynomials. [ABSTRACT FROM PUBLISHER]

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]

Cellular automata-based bit error correcting codes over binary field was originally studied by Chowdhury et al. (IEEE Trans. Comput. 43:759-764, ) and also an algorithm for decoding such codes was introduced. Further, for the binary field case, it was shown that cellular automata-based error correcting codes have faster decoding algorithm than the classical linear syndrome decoding algorithm. We generalize Chowdhury's approach from binary to primitive finite fields and we also compare the classical syndrome decoding with the one introduced in this work. We show that error correcting codes obtained via cellular automata have faster decoding than the classical ones. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of linear codes over the non chain ring $$Z_{3}[v]/\langle v^{3}-v\rangle $$ . In order to study the codes, we first study the structure of this ring via a distance preserving Gray map which also induces a relation between codes over this ring and ternary codes. Further, the algebraic structure of cyclic and dual codes is also studied. A MacWilliams type Identity between the Gray weight enumerators of the original code and its dual is established. In all cases examples that illustrate the theorems and lemmas are provided. Also, a BCH-type bound and an example that attains this bound is presented. [ABSTRACT FROM AUTHOR]

In this paper, we obtain bounds for error correcting codes of some particular homogeneous weights over the ring of integers modulo $${2^l( {l\ge 3})}$$. We also study these bounds to check for perfect linear codes and have shown the non-existence of one error correcting perfect codes with respect to the homogeneous weight. [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]

The reversibility problem for linear cellular automata with null boundary defined by a rule matrix in the form of a pentadiagonal matrix was studied over the binary field ℤ2 by Martín del Rey et al. [ Appl. Math. Comput. 217, 8360 (2011)]. Recently, the reversibility problem of 1D Cellular automata with periodic boundary has been extended to ternary fields and further to finite primitive fields ℤp by Cinkir et al. [ J. Stat. Phys. 143, 807 (2011)]. In this work, we restudy some of the work done in Cinkir et al. [ J. Stat. Phys. 143, 807 (2011)] by using a different approach which is based on the theory of error correcting codes. While we reestablish some of the theorems already presented in Cinkir et al. [ J. Stat. Phys. 143, 807 (2011)], we further extend the results to more general cases. Also, a conjecture that is left open in Cinkir et al. [ J. Stat. Phys. 143, 807 (2011)] is also solved here. We conclude by presenting an application to Error Correcting Codes (ECCs) where reversibility of cellular automata is crucial. [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]

New methods of teaching linear algebra in the undergraduate curriculum have attracted much interest lately. Most of this work is focused on evaluating and discussing the integration of special computer software into the Linear Algebra curriculum. In this article, I discuss my approach on introducing the concept of eigenvectors and eigenvalues, which I have used for the last 3 years in my Linear Algebra course. I offer some examples on how I have attracted the interest of our students via Hill ciphering, a method of cryptography. After emphasizing the effect of a linear transformation in a vector space and the importance of eigenvectors, I show how students’ motivation and understanding towards one of the abstract concepts in Linear Algebra; eigenvalues and eigenvectors have grown positively. [ABSTRACT FROM PUBLISHER]

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 study we determine the structure of reversible DNA codes obtained from skew cyclic codes. We show that the generators of such DNA codes enjoy some special properties. We study the structural properties of such family of codes and we also illustrate our results with examples. [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 paper, we will study the structure of ℤ 4 R -additive codes where ℤ 4 = { 0 , 1 , 2 , 3 } is the well-known ring of 4 elements and R is the ring given by R = ℤ 4 + u ℤ 4 + v ℤ 4 , where u 2 = u , v 2 = v and u v = v u = 0. We will classify all maximum distance separable codes with respect to the Singleton bound (MDSS) over ℤ 4 R. Then we will focus on ℤ 4 R -additive cyclic and constacyclic codes. We will find a unique set of generator polynomials for these codes and determine minimum spanning sets for them. We will also find the generator polynomials for the dual of any ℤ 4 R -additive cyclic or constacyclic code. [ABSTRACT FROM AUTHOR]

In this paper, a new class of additive codes which is referred to as ℤ2ℤ2[u]-additive codes is introduced. This is a generalization towards another direction of recently introduced ℤ2ℤ4-additive codes [J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rif´a, and M. Villanueva, ℤ2ℤ4-linear codes: Generator matrices and duality, Designs Codes Cryptogr. 54(2) (2010), pp. 167–179]. ℤ2ℤ4-additive codes have shown to provide a promising class of codes with their algebraic structure and applications such as steganography. The standard generator matrices are established and by introducing orthogonality the parity-check matrices are also obtained. A MacWilliams-type identity that relates the weight enumerator of a code with its dual is proved. Furthermore, a Gray map that maps these codes to binary codes is defined and some examples of optimal codes which are the binary Gray images of ℤ2ℤ2[u]-additive codes are presented. [ABSTRACT FROM PUBLISHER]

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 paper, we study one dimensional finite linear cellular automata with reflective boundary condition by using matrix algebra built on the field ℤp. We present an algorithm for determining the reversibility of this family of cellular automata. We also answer the reversibility question for some special subfamilies. Finally, we present some examples of this family of cellular automata under the reflective boundary condition. [ABSTRACT FROM AUTHOR]

The reversibility problem for linear cellular automata with null boundary defined by a rule matrix in the form of a pentadiagonal matrix was studied recently over the binary field ℤ (del Rey and Rodriguez Sánchez in Appl. Math. Comput., , doi:). In this paper, we study one-dimensional linear cellular automata with periodic boundary conditions over any finite field ℤ. For any given p≥2, we show that the reversibility problem can be reduced to solving a recurrence relation depending on the number of cells and the coefficients of the local rules defining the one-dimensional linear cellular automata. More specifically, for any given values (from any fixed field ℤ) of the coefficients of the local rules, we outline a computer algorithm determining the recurrence relation which can be solved by testing reversibility of the cellular automaton for some finite number of cells. As an example, we give the full criteria for the reversibility of the one-dimensional linear cellular automata over the fields ℤ and ℤ. [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]