Irfan Siap

• Ph.D.
• Professor at American University of Sharjah

https://www.sciencedirect.com/science/article/abs/pii/S1071579725000218

Let p be a prime integer and Fp the finite field of order p. Moreover, let q1 and q2 be prime integers such that p is a quadratic residue modulo q1 and q2. In this paper, we will introduce the class of double quadratic residue (QR) codes of length n=q1+q2, over the ring Fp×Fp. To study these codes and their properties, we will focus on two cases: I...

In this paper, we will study the structure of [Formula: see text]-additive codes where [Formula: see text] is the well-known ring of 4 elements and [Formula: see text] is the ring given by [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text]. We will classify all maximum distance separable codes with respect t...

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...

In this study we determine the structure of m-adic residue codes over the non-chain ring Fq[v]/(v 2 − v) and present some promising examples of such codes that have optimal parameters with respect to Griesmer Bound. Further, we show that the generators of m-adic residue codes serve as a natural and suitable application for generating reversible DNA...

In this study we determine the structure of m-adic residue codes over the non-chain ring Fq[v]/(v² − v) and present some promising examples of such codes that have optimal parameters with respect to Griesmer Bound. Further, we show that the generators of m-adic residue codes serve as a natural and suitable application for generating reversible DNA...

Recently some special type of mixed alphabet codes that generalize the standard codes has attracted much attention. Besides Z2Z4-additive codes, Z2Z2[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 Z2Z2[u3] where Z2[u3]={0,1,u,1+u,u2,1+u...

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.

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 , and by using a form...

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...

In this work, we determine self dual and self orthogonal codes arising from constacyclic codes over group algebras. Also, based on these codes we obtain some good parameters for quantum error-correcting codes.

Constacyclic codes are preferred in engineering applications due to their efficient encoding process via shift registers. The class of constacyclic codes contains cyclic and negacyclic codes. The relation and presentation of cyclic codes as group algebras has been considered. Here for the first time, we establish a relation between constacyclic cod...

In this paper we study the structure of specific linear codes called DNA codes. The first attempts on studying such codes have been proposed over four element rings which are naturally matched with DNA four letters. Later, double (pair) DNA strings or in general $k$-DNA strings called $k$-mers have been matched with some special rings and codes ove...

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.

In this paper we study the structure of specific linear codes called DNA codes. The first attempts on studying such codes have been proposed over four element rings which are naturally matched with DNA four letters. Later, double (pair) DNA strings or in general $k$-DNA strings called $k$-mers have been matched with some special rings and codes ove...

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...

In this paper, we extend the results given in [3] to a nonchain ring Rp = Fp + vFp + … + vp−1Fp, where vp = v and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring Rp and study the structure of their duals. We classify cyclic codes containing their duals over Rp by giving necessary and sufficient conditi...

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...

Following the very recent studies on ℤ <sub xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ℤ <sub xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -additive codes, ℤ <sub xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ℤ <sub xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> [u]-linear codes have been introduced by Aydogdu et al. In this...

In this paper, we determine self dual and self orthogonal codes arising from negacyclic codes over the group ring (Fq + υFq) G. By taking a suitable Gray image of these codes we obtain many good parameter quantum error-correcting codes over Fq.

Cellular automata have rich computational properties and provide many models in mathematical and physical processes. In this work, one of the most commonly used neighborhood types of two dimensional (2D) cellular automata which is called Moore neighborhood in two dimensional integer lattice is considered. We study the characterization of 2D linear...

In this paper the reversibility problem of a family of two-dimensional cellular automata is completely resolved. It is well known that the reversibility problem is a very difficult one in general. In order to determine whether a cellular automaton is reversible or not the reversibility of its rule matrix is studied via linear algebraic tools. Howev...

In this study, we introduce a new Gray map which preserves the orthogonality from the chain ring F-2 [u] / (u(s)) to F-2(s) where F-2 is the finite field with two elements. We also give a condition of the existence for cyclic codes of odd length containing its dual over the ring F-2 [u] / (u(s)). By taking advantage of this Gray map and the structu...

Lately, quantum error correcting codes have been considered as the Gray images of some special codes over various rings [1], [4], [9]. In this study, we investigate the structure of the cyclic codes of arbitrary length over the finite nonchain ring Rp = Fp+vFp+···+vp-1Fp where vp = v and obtain quantum error correcting codes over Fp by using the Gr...

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 consta...

We study theoretical structure and classification of two-dimensional (2D) 3-states uniform cellular automata (CA) based on their visual behaviors. Although the basics of a CA is a discrete dynamic structure and modelled locally, the behavior at large times and spatial scales could be a close to a continuous system. Using some basic properties, it c...

This paper determines the structures of one-homogeneous weight codes over finite chain rings and studies the algebraic properties of these codes. We present explicit constructions of one-homogeneous weight codes over finite chain rings. By taking advantage of the distance- preserving Gray map defined in [7] from the finite chain ring to its residue...

In this paper, a new class of additive codes which is referred to as ℤ2 ℤ2[u]-additive codes is introduced. This is a generalization towards another direction of recently introduced ℤ2 ℤ4-additive codes [J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rif´a, and M. Villanueva, ℤ2 ℤ4-linear codes: Generator matrices and duality, Designs Codes Cryptogr...

In this paper, we generalize the lifted polynomials which generate reversible codes over Fq, a finite field with q element. 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] over F16. Lifted polynomials have proven to be very advantageo...

In this paper, we study the algebraic structure of -additive codes which are -submodules where is prime, and and are positive integers. -additive codes naturally generalize and -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 determ...

In this paper quadratic residue codes over the ring Fp+vFp 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 p= 2, Euclidean and Hermitian self-dual families of codes as extended quad...

In this work, quadratic residue codes over the ring F2+uF2+u2F2F2+uF2+u2F2 with u3=uu3=u are considered. A duality and distance preserving Gray map from F2+uF2+u2F2F2+uF2+u2F2 to F23 is defined. By using quadratic double circulant, quadratic bordered double circulant constructions and their extensions self-dual codes of different lengths are obtain...

n this correspondence, we consider quadratic double and bordered double circulant construction methods over the ring R := F2 + uF2 + u2F2, where u3 = 1. Among other examples, extremal binary self-dual codes of length 66 are obtained by these constructions. These are extended by using extension theorems for self-dual codes and as a result 8 new extr...

In this study we focus on codes over a special family of commutative rings where we are able to construct a map that gives a correspondence between k-bases (k-letter words) of DNA with elements of the ring. By making use of so called coterm polynomials, we are able to solve the reversibility and complement problems in DNA codes and construct DNA co...

In this work, we study the structure of skew constacyclic codes over the ring R = F 4[v]/〈v 2 − v〉 which is a non chain ring with 16 elements where F 4 denotes the field with 4 elements and v an indeterminate. We relate linear codes over R to codes over F 4 by defining a Gray map between R and \(F_{4}^{2}.\) Next, the structure of all skew constacy...

In this paper skew cyclic codes over the the family of rings Fq+vFq with v2 = v are studied for the �rst time in its generality. Structural properties of skew cyclic codes over Fq + vFq are investigated through a decomposition theorem. It is shown that skew cyclic codes over this ring are principally generated. The idempotent generators of skew-cyc...

In this work, quadratic double and quadratic bordered double circulant constructions are applied to F_4 + uF_4 as well as F_4, as a result of which 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 7 extremal self-dual binary codes of length 58, 2...

Cellular automata-based bit error correcting codes over binary field was originally studied by Chowdhury et al. (IEEE Trans. Comput. 43:759–764, 1994) 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...

In this study, we consider linear and especially cyclic codes over the non-chain ring $Z_{p}[v]/\langle v^{p}-v\rangle$ where $p$ is a prime. This is a generalization of the case $p=3.$ Further, in this work the structure of constacyclic codes are studied as well. This study takes advantage mainly from a Gray map which preserves the distance betwee...

In this paper, we study Z(2)Z(4)-additive cyclic codes. These codes are identified as Z(4)[x]-submodules of the ring R-r,R-s = Z(2)[x]/< x(r) - 1 > x Z(4) [x]/< x(s)-1 >. The algebraic structure of this family of codes is studied and a set of generator polynomials for this family as a Z(4)[x]-submodule of the ring R-r,R-s is determined. We show tha...

Let n = α + 2β. In this paper, we introduce a new type of linear and cyclic codes defined over the ring ℤ2R where ℤ2 = {0;1} is the binary finite field and the ring R = {0;1;u;u + 1} where u2 = 0: We give the definition of these codes as subsets of the ring ℤ2α × Rβ: We give a one-to-one correspondence between elements in ℤ2α × Rβ and elements in t...

A family of one-dimensional finite linear cellular automata with reflective boundary condition over the field Zp is defined. The generalizations are the radius and the field that states take values. Here, we establish a connection between reversibility of cellular automata and the rule matrix of the cellular automata with radius three. Also, we pro...

This paper investigates the theoretical aspects of two-dimensional linear cellular automata with image appli-cations. We consider geometrical and visual aspects of patterns generated by cellular automata evolution. The present work focuses on the theory of two-dimensional linear cellular automata with respect to uniform periodic and adiabatic bound...

In this work, we study the structure of two-dimensional linear hybrid cellular automata with respect to adiabatic boundary condition. Further, we check the performance of hybrid cellular automata constructed through the members of this family in generating pseudo random bits.

In this work, the main purpose is to extend some well known binary and quaternary codes to the ring F2+uF2. Reed-Muller, Goethals, Delsarte-Goethals codes are extended, their properties and relations to binary and quaternary versions are studied. Double error correcting families of codes as Goethals and shortened Goethals codes over F2+u F2 are als...

In this work, the main purpose is to extend some well known binary and quaternary codes to the ring F2+uF2F2+uF2. Reed–Muller, Goethals, Delsarte–Goethals codes are extended, their properties and relations to binary and quaternary versions are studied. Double error correcting families of codes as Goethals and shortened Goethals codes over F2+uF2F2+...

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,...

In this paper quadratic residue codes over the ring Fp+vFpFp+vFp 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 p=2p=2, Euclidean and Hermitian self-dual families of codes as exten...

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 d...

ℤ2ℤ4-additive codes, as a special class of abelian codes, have found a very welcoming place in the recent studies of algebraic coding theory. This family in one hand is similar to binary codes on the other hand is similar to quaternary codes. The structure of ℤ2ℤ4- additive codes and their duals has been determined lately. In this study we investig...

In this paper, we introduce a family of one dimensional finite linear cellular automata with periodic boundary condition over primitive finite fields with p elements (Zp) which leads to a generalization in two directions: the radius and the field that states take values. This family of cellular automata is called (2r + 1)-cyclic cellular automata s...

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.

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 charact...

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...

In this paper, we count the number of matrices whose rows generate different $\mathbb{Z}_2\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the number of matrices whose rows generate vector spaces with particular dimension over finite fields. Due to this similarity we name this numbers as M...

In this paper, we study 2-dimensional finite cellular automata defined by hexagonal local rule with periodic boundary over the field Z3. We construct the rule matrix corresponding to the hexagonal cellular automata. For some given coeficients and the number of columns of hexagonal information matrix, we prove that the hexagonal cellular automata ar...

In this paper, we introduce a new family of polynomials which generates reversible codes over a finite field with sixteen elements (F-16 or G Gamma(16)). We name the polynomials in this family as lifted polynomials. Some advantages of lifted polynomials are that they are easy to construct, there are plenty of examples of them and it is easy to dete...

The m-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). m-spotty byte error control codes are very suitable for burst correction. Here, we introduce the m-spotty weights and m-spotty weight enumerator of...

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...

In this work we study the algebraic behavior of three dimensional linear cellular automata over Zm: we provide necessary and sufficient conditions for a three dimensional linear cellular automata over Zm to be reversible or irreversible. As a consequence of our result we characterize three dimensional linear cellular automata under the null boundar...

Cellular automata are simple mathematical representation of complex dynamical systems. Therefore there are several applications of cellular automata in many areas such as coding, cryptography, VLSI design etc. [1, 2]. In this study, a recurrence relation for computation minimal polynomial of rule matrix of linear elementary rule 150 with reflectiv...

Reed-Solomon codes are very convenient for burst error correction which occurs frequently in applications, but as the number of errors increase, the circuit structure of implementing Reed-Solomon codes becomes very complex. An alternative solution to this problem is the modular and regular structure of cellular automata which can be constructed wit...

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 exam...

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...

In this paper, we study quasi-cyclic codes over the ring R=F2+uF2={0,1,u,u+1}R=F2+uF2={0,1,u,u+1} where u2=0u2=0. By exploring their structure, we determine the type of one generator quasi-cyclic codes over R and the size by giving a minimal spanning set. We also determine the rank and introduce a lower bound for the minimum distance of free quasi-...

In this paper, we define a family of Generalized Gaussian Numbers that gives the number of linear codes over Galois rings directly. Also, we study some of their properties and obtain some relations between them.

We establish some new properties and identities of Generalized Gaussian Numbers (GGN) which are de�ned recently in [10, 11] parallel to those of Gaussian coe�cients. We present generating functions and some properties which are very useful for GGN. We obtain some family of sequences which are unimodal and present the log-concavity property of GGN....

In this paper, we study a 2-dimensional cellular automaton generated by a new local rule of the nearest neighborhoods and the next nearest neighborhoods under periodic boundary conditions over the field 3. We obtain the rule matrix of this cellular automaton and characterize the cellular automaton with some of its important characteristics. We get...

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 b...

We propose and test two dimensional linear hybrid cellular automata with Moore neighborhood instead of linear Wolfram cellular automata to determine whether it is convenient to using as a pseudo random number generator. The preliminary results show significance of generating pseudorandom numbers. These hybrid cellular automata have passed the well-...

In this work we introduce and study a new family of one dimensional nonlinear cellular automaton which we name as quadratic cellular automata over ternary fields ( 3 Z ). This family is defined by using the quadratic forms as local transition functions. Further, we define hybrid quadratic cellular automata. Under periodic, null, and reflective boun...

The set of papers [3], [4], [6] and [7] (Chattopadhyay et al., 1999; Dihidar and Choudhury, 2004; Khan et al., 1997, 1999) deals with the behavior of the uniform two-dimensional cellular automata over binary fields (Z2). Some structural properties and precise mathematical models using matrix algebra over the field Z2 are reported for characterizing...

A characterization of von Neumann neighborhood cellular automata was extensively studied in the literature. In this paper, we study one of the most commonly used neighborhood types of CA which is called Moore neighborhood in two dimensional lattice. We investigate the characterization of two dimensional Moore neighborhood cellular automata over ter...

We study quaternary quasi-cyclic (QC) codes with even-length components. We determine the structure of one-generator quaternary QC codes whose cyclic components have even length. Making use of their structure, we establish the size of these codes and give a lower bound for their minimal distance. We present some examples of codes from this family w...

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 ℤ2 (del Rey and Rodriguez Sánchez in Appl. Math. Comput., 2011, doi:10.1016/j.amc.2011.03.033). In this paper, we study one-dimensional linear cellular automata with peri...

In this paper we study a special type of linear codes, called skew cyclic codes, in the most general case. This set of codes is a gen- eralization of cyclic codes but constructed using a non-commutative ring called the skew polynomial ring. In previous works these codes have been studied with certain restrictions on their length. This work examines...

2-dimensional finite cellular automata defined by local rule based on hexagonal cell structure are studied. Rule matrix of the hexagonal finite cellular automaton is obtained. The rank of rule matrices related to hexagonal finite cellular automata via an algorithm is computed. By using the matrix algebra it is shown that the hexagonal finite cellul...

An important problem in cellular automata theory is the reversibility of a cellular automaton which is related to the existence of Garden of Eden configurations in cellular automata. In this paper, we study new local rules for two-dimensional cellular automata over the ternary field Z3 (the set of integers modulo three) with some of their important...

In this work, the cardinality of the minimal R-covers of finite rings with respect to the RT-metric is established. By generalizing the result in Nakaoka and dos Santos (2010) [1], the minimal cardinalities of 0-short coverings of finite chain rings are calculated. The connection between R-short coverings of rings with respect to the RT-metric and...

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 ;¿]. After a brief description of the skew polynomial ring F [ x ;¿], it is shown that skew QC codes are left submodules of the ring Rsl =( F [ x ;¿]/( xs -1) )...

It is a well established fact that m-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 m-spotty weight enumerator of a bin...

An important problem in cellular automata theory is the reversibility of a cellular automaton which is related to the existence of Garden of Eden configurations in cellular automata. In this paper, we study new local rules for two-dimensional cellular automata over the ternary field Z3 (the set of integers modulo three) with some of their important...

Enumerating burst errors enables to obtain bounds on parameters of codes. Recently, S. Jain [Linear Algebra Appl. 418, No. 1, 130–141 (2006; Zbl 1103.94026)] established a Reiger’s type bound for burst error correcting matrix codes over finite fields with respect to a non Hamming metric. Here, we extend these results to array codes over finite ring...

We study the structure of cyclic DNA codes over the ring F2[u]/(u2-1). We employ the deletion similarity distance on the set of codewords. A set of generators for this type of codes is found. We also study the CG-content of these type of codes and their deletion distance. Examples of cyclic DNA codes are constructed with their CG-content and their...

We study the structure of (1+u)-constacyclic codes of an arbitrary length n over the ring F 2 +uF 2 . We find a set of generators for each (1+u)-constacyclic code and its dual. We study the rank of cyclic codes and find their minimal spanning sets. We prove that the Gray image of a (1+u)-constacyclic code is a binary cyclic code of length 2n. We co...

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 non-commutative ring called the skew polynomial rings $F[x;\theta ]$. After a brief description of the skew polynomial ring $F[x;\theta ]$ it is shown that skew QC codes are left submodules of the ring $R_{s}^{l}=(F[x;\the...

In this paper we construct a special type of cyclic codes called cyclic DNA codes over the ring R = {0, 1, u, u + 1} where u² = 1. These codes are important in the subject of biomolecular computation and DNA computing. We apply a more suitable measure (the similarity measure) than the Hamming distance measure on the set of codewords. Our...

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, whic...