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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.

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]

Secret sharing scheme is an efficient method to hide secret key or secret image by partitioning it into parts such that some predetermined subsets of partitions can recover the secret but remaining subsets cannot. In 1979, the pioneer construction on this area was given by Shamir and Blakley independently. After these initial studies, Asmuth-Bloom and Mignotte have proposed a different (k; n) threshold modular secret sharing scheme by using the Chinese remainder theorem. In this study, we explore the generalization of Mignotte's scheme to Euclidean domains for which we obtain some promising results. Next, we propose new algorithms to construct threshold secret image sharing schemes by using Mignotte's scheme over polynomial rings. Finally, we compare our proposed scheme to the existing ones and we show that this new method is more efficient and it has higher security. [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]

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 focus on reversible cyclic codes which correspond to reversible DNA codes or reversible-complement DNA codes over a family of finite chain rings, in an effort to extend what was done by Yildiz and Siap in [20]. The ring family that we have considered are of size 22k , k = 1, 2, . . . and we match each ring element with a DNA 2k-1-mer. We use the so-called u2-adic digit system to solve the reversibility problem and we characterize cyclic codes that correspond to reversible-complement DNA-codes. We then conclude our study with some examples. [ABSTRACT FROM AUTHOR]

Secret sharing schemes (SSS) offer efficient methods in protecting a secret data shared in parts by shareholders against unauthorized people. Furthermore, this sharing is formed in a way that only a predesigned minimum number of shareholders can solve the secret by using their shares. Many methods have been developed over the last decades. In this work, we present a novel ( l , t + l ) threshold secret sharing scheme by using error correcting codes. The security of this scheme is based on the hardness of syndrome decoding problem. [ABSTRACT FROM AUTHOR]

In this work, we study the structure of linear, constacyclic and cyclic codes over the ring $$R=F_{4}[v]/(v^{2}-v)$$ and establish relations to codes over $$ F_{4}$$ by defining a Gray map between R and $$F_{4}^{2}$$ where $$F_4$$ is the field with 4 elements. Constacyclic codes over R are shown to be principal ideals. Further, we study skew constacyclic codes over R. The structure of all skew constacyclic codes is completely determined. Furthermore, we introduce reversible codes which provide a rich source for DNA codes. We conclude the paper by obtaining some DNA codes over R that attain the Griesmer bound. [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, 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 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]

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 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 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]

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]

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]

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]

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]