Nuh Aydin

• PhD
• Professor at Kenyon College

The paper introduces a new class of linear codes called (\sigma,\bar{a})-polycyclic codes. By examining their algebraic structure, we obtain certain basic properties of these codes. We show that (\sigma,\bar{a})-polycyclic codes generalize some of the well known classes of linear codes. Further, we describe the class of -generator generalized quasi...

In this paper, we propose a novel hybrid encryption algorithm that enhances security by integrating DNA encoding, the Rabin algorithm, a one-time pad (OTP), and a Feistel-inspired structure. The algorithm begins with the application of a DNA-based OTP key, used exclusively for a single secure communication session. In the second step, it combines p...

This paper explores different types of skew cyclic codes by generating special subclasses with additional desirable properties. Specifically, we are interested in skew cyclic codes over mixed rings. We study some algebraic and structural properties of these codes and their constructions. We study skew cyclic codes over the mixed alphabet ring $\mat...

Recently, many good quantum codes over various finite fields $F_q$ have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from cyclic codes or their generalizations over $F_q$. Unless explicit benefits are demonstrated for the indir...

In this paper we generalize the notion of n-isometry and n-equivalence relation introduced by Chen et al. in [13], [12] to classify constacyclic codes of length n over a finite field , where is a prime power, to the case of skew constacyclic codes without derivation. We call these relations respectively -equivalence and -isometric relation, where n...

This paper deals with linear codes invariant under an endomorphism $T,$ that we call $T$-codes. Since any endomorphism can be represented by a matrix when a basis is fixed, we shall, for simplicity, consider linear codes invariant under right multiplication by a square matrix $M \in \mathbb{M}_{n}(\Fq)$, and we call them $M$-codes. In particular, w...

In this paper, we are interested in right (resp., left) quasi-polycyclic (QP) codes of length $n=m\ell$ with an associated vector $a=(a_0,a_1,\ldots, a_{m-1})\in \mathbb{F}_{q}^{^m}$, which are a generalization of quasi-cyclic codes (QC) and quasi-twisted (QT) codes. They are defined as invariant subspaces of $\mathbb{F}_{_q}^{^{n}} $ by the right...

In this paper we study the structure and properties of additive right and left polycyclic codes induced by a nonbinary vector \(\textbf{a}\in \mathbb {F} _{4}^{n}\), where \(\mathbb {F}_{4}\) is the finite field of order 4. We show that additive right and left polycyclic codes are \(\mathbb {F}_{2}[x]\) -submodules of the rings \(R_{n}=\mathbb {F}_...

In this paper, we study linear codes invariant under a cyclic endomorphism T, called T-cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix...

Determining the best possible values of the parameters of a linear code is one of the most fundamental and challenging problems in coding theory. There exist databases of best-known linear codes (BKLC) over small finite fields. In this work, we establish a database of BKLCs over the field GF(17) together with upper bounds on the minimum distances f...

In this paper, we are interested in linear codes invariant under the $(\sigma,\delta)$-action of a given matrix $M,$ that we call $(M,\sigma,\delta)$-skew codes. Various subclasses of these codes, depending on the $(\sigma,\delta)$-cyclical property of $M,$ are derived. In particular, we distinguish between three fundamental types: $(M,\sigma,\delt...

Information security is a crucial need in the modern world. Data security is a real concern, and many customers and organizations need to protect their sensitive information from unauthorized parties and attackers. In previous years, numerous cryptographic schemes have been proposed. DNA cryptography is a new and developing field that combines the...

In this paper, we study the skew-cyclic codes (also called \(\varvec{\theta }\)-cyclic codes) over the ring \(\varvec{S}=\varvec{\mathbb {Z}}_{{\textbf {4}}}+\varvec{u}\mathbb {Z}_{{\textbf {4}}}+v\mathbb {Z}_{{\textbf {4}}}\), where \(\varvec{u}^2=v^{{\textbf {2}}}=\varvec{u}\varvec{v}=\varvec{v}\varvec{u}={\textbf {0}}\). Some structural properti...

Research on codes over finite rings has intensified after the discovery that some of the best binary nonlinear codes can be obtained as images of \(\mathbb {Z}_4\)-linear codes. Codes over various finite rings have been a subject of much research in coding theory after this discovery. Many of these rings are extensions of \(\mathbb {Z}_4\) and nume...

In the current paper, we study skew cyclic codes over the ring \({\mathcal {S}}:={\mathbb {F}}_2 \times ({\mathbb {F}}_2+v{\mathbb {F}}_2)\) with \(v^2=v\). We investigate the structural properties of skew cyclic codes over the ring \({\mathcal {S}}\). Also, we describe the dual codes of skew cyclic codes with respect to the Euclidean inner product...

One of the most important and challenging problems in coding theory is to construct codes with best possible parameters and properties. The class of quasi-cyclic (QC) codes is known to be fertile to produce such codes. Focusing on QC codes over the binary field, we have found 113 binary QC codes that are new among the class of QC codes using an imp...

Multi-twisted (MT) codes were introduced as a generalization of quasi-twisted (QT) codes. QT codes have been known to contain many good codes. In this work, we show that codes with good parameters and desirable properties can be obtained from MT codes. These include best known and optimal classical codes with additional properties such as reversibi...

In this paper, we investigate the structure and properties of skew negacyclic codes and skew quasi-negacyclic codes over the ring [Formula: see text] Some structural properties of [Formula: see text] are discussed, where [Formula: see text] is an automorphism of [Formula: see text] A skew quasi-negacyclic code of length [Formula: see text] with ind...

Recently, a new algorithm to test equivalence of two cyclic codes has been introduced which is efficient and produced useful results. In this work, we generalize this algorithm to constacyclic codes. As an application of the algorithm we found many constacyclic codes with good parameters and properties. In particular, we found 22 new codes that imp...

Research on codes over finite rings has intensified since the discovery in 1994 of the fact that some best binary non-linear codes can be obtained as images of $\mathbb{Z}_4$-linear codes. Codes over many different finite rings has been a subject of much research in coding theory after this discovery. Many of these rings are extensions of $\mathbb{...

Polycyclic codes are a generalization of cyclic and constacyclic codes. Even though they have been known since 1972 and received some attention more recently, there have not been many studies on polycyclic codes. This paper presents an in-depth investigation of polycyclic codes associated with trinomials. Our results include a number of facts about...

In this paper, we study the structure and properties of additive right and left polycyclic codes induced by a binary vector \begin{document}$ a $\end{document} in \begin{document}$ \mathbb{F}_{2}^{n}. $\end{document} We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, w...

Convolutional codes are error-correcting linear codes that utilize shift registers to encode. These codes have an arbitrary block size and they can incorporate both past and current information bits. DNA codes represent DNA sequences and are defined as sets of words comprised of the alphabet A, C, T, G satisfying certain mathematical bounds and con...

Cyclic codes are among the most important families of codes in coding theory for both theoretical and practical reasons. Despite their prominence and intensive research on cyclic codes for over a half century, there are still open problems related to cyclic codes. In this work, we use recent results on the equivalence of cyclic codes to create a mo...

In this paper we study the structure and properties of additive right and left polycyclic codes induced by a binary vector $a$ in $\mathbb{F}_{2}^{n}.$ We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if $C$ is a right polycyclic code induced by a vector...

One of the main goals of coding theory is to construct codes with best possible parameters and properties. A special class of codes called quasi-twisted (QT) codes is well-known to produce codes with good parameters. Most of the work on QT codes has been over the 1-generator case. In this work, we focus on 2-generator QT codes and generalize the AS...

Recently, a new algorithm to test equivalence of two cyclic codes has been introduced which is efficient and produced useful results. In this work, we generalize this algorithm to constacyclic codes. As an application of the algorithm we found many constacyclic codes with good parameters and properties. In particular, we found 23 new codes that imp...

One of the most important and challenging problems in coding theory is to construct codes with best possible parameters and properties. The class of quasi-cyclic (QC) codes is known to be fertile to produce such codes. Focusing on QC codes over the binary field, we have found 113 binary QC codes that are new among the class of QC codes using an imp...

Quantum error correcting codes (QECC) is becoming an increasingly important branch of coding theory. For classical block codes, a \href{codetables.de} {comprehensive database of best known codes} exists which is available online at \cite{codetables}. The same database contains data on best known quantum codes as well, but only for the binary field....

Cyclic codes are among the most important families of codes in coding theory for both theoretical and practical reasons. Despite their prominence and intensive research on cyclic codes for over a half century, there are still open problems related to cyclic codes. In this work, we use recent results on the equivalence of cyclic codes to create a mo...

Polycyclic codes are a generalization of cyclic and constacyclic codes. Even though they have been known since 1972 and received some attention more recently, there have not been many studies on polycyclic codes. This paper presents an in-depth investigation of polycyclic codes associated with trinomials. Our results include a number of facts about...

In this paper, we introduce skew cyclic codes over the mixed alphabet [Formula: see text], where [Formula: see text] is the finite field with 4 elements and [Formula: see text]. Our results include a description of the generator polynomials of such codes and a necessary and sufficient condition for an [Formula: see text]-skew cyclic code to be reve...

We introduce skew cyclic codes over the finite ring $\R$, where $u^{2}=0,v^{2}=v,w^{2}=w,uv=vu,uw=wu,vw=wv$ and use them to construct reversible DNA codes. The 4-mers are matched with the elements of this ring. The reversibility problem for DNA 4-bases is solved and some examples are provided.

This paper considers a new alphabet set, which is a ring that we call 𝔽4R, to construct linear error-control codes. Skew cyclic codes over this ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize 𝔽4R-skew cyc...

It contains an introduction and nine chapters that are divided into sections. The introduction contains the definition of area and the terminology used in it.

Multi-twisted (MT) codes were introduced as a generalization of quasi-twisted (QT) codes. QT codes have been known to contain many good codes. In this work, we show that codes with good parameters and desirable properties can be obtained from MT codes. These include best known and optimal classical codes with additional properties such as reversibi...

One of the most important and challenging problems in coding theory is to determine the optimal values of the parameters of a linear code and to explicitly construct codes with optimal parameters, or as close to the optimal values as possible. The class of quasi-twisted (QT) codes has been very promising in this regard. Over the past few decades va...

In this paper, we study the algebraic structure of additive cyclic codes over the alphabet \({\mathbb {F}}_{2}^{r}\times {\mathbb {F}}_{4}^{s}={ \mathbb {F}}_{2}^{r}{\mathbb {F}}_{4}^{s},\) where r and s are non-negative integers, \(\mathbb {F}_{2}={\mathbb {GF}}(2)\) and \(\mathbb {F}_{4}={\mathbb {GF}} (4)\) are the finite fields of 2 and 4 eleme...

In this paper, we study skew constacyclic codes over the ring ZqR where R = Zq + uZq, q = p s for a prime p and u 2 = 0. We give the definition of these codes as subsets of the ring Z α q R β. Some structural properties of the skew polynomial ring R[x, Θ] are discussed, where Θ is an automorphism of R. We describe the generator polynomials of skew...

We introduce skew cyclic and skew constacyclic codes over the local Frobenius non-chain rings of order 16 by defining non-trivial automorphisms on these rings. We study the Gray images of these codes, obtaining a number of binary and quaternary codes with good parameters as images of skew cyclic codes over these rings.

Jamshīd al-Kāshī’s Miftāḥ al-Ḥisab (Key to Arithmetic) was largely unknown to researchers until the mid-20th century, and has not been translated to English until now. This is the second book in a multi-volume set that finally brings al-Kāshī’s groundbreaking textbook to English audiences in its entirety. As soon as it was studied by modern researc...

A fundamental problem in coding theory is the explicit construction of linear codes with best possible parameters. A search algorithm (ASR) on certain types of quasi-twisted (QT) codes has been very fruitful to address this challenging problem. In this work, we generalize the ASR algorithm to make it more comprehensive. The generalization is based...

One of the most important and challenging problems in coding theory is to explicitly construct linear codes with best possible parameters. Computers are often used to search for optimal codes. However, given the large size of the search space and computational complexity of determining the minimum distance, researchers usually focus on promising cl...

One of the main problems of coding theory is to construct codes with best possible parameters. Cyclic codes and their various generalizations, such as quasi-twisted codes, have been a fruitful source in achieving this goal. Recently, a new generalization of cyclic codes that are known as skew cyclic codes have been introduced and some new codes obt...

In the Name of God, Most Gracious Most merciful. Lord make it easy, not hard.

Explicit construction of linear codes with best possible parameters is one of the major and challenging problems in coding theory. Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, are known to contain many codes with best known parameters. Despite the fact that these classes of codes have been extensively searched,...

Jamshīd al-Kāshī’s Miftāḥ al-Ḥisab (Key to Arithmetic) was largely unknown to researchers until the mid-20th century, and has not been translated to English until now. This book begins a multi-volume set that finally brings al-Kāshī’s groundbreaking textbook to English audiences in its entirety. As soon as it was studied by modern researchers, it c...

In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0$. We give the definition of these codes as subsets of the ring $\mathbb{Z}_{q}^{\alpha}R^{\beta}$. Some structural properties of the skew polynomial ring $ R[x,\theta]$ are discussed, where...

In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0$. We give the definition of these codes as subsets of the ring $\mathbb{Z}_{q}^{\alpha}R^{\beta}$. Some structural properties of the skew polynomial ring $ R[x,\theta]$ are discussed, where...

In this paper, we study skew cyclic codes with arbitrary length over the ring $R=\mathbb{F}_{p}+u\mathbb{F}_{p}$ where $p$ is an odd prime and $% u^{2}=0$. We characterize all skew cyclic codes of length $n$ as left $% R[x;\theta ]$-submodules of $R_{n}=R[x;\theta ]/\langle x^{n}-1\rangle $. We find all generator polynomials for these codes and des...

Explicit construction of linear codes with best possible parameters is one of the major problems in coding theory. Among all alphabets of interest, the binary alphabet is the most important one. In this work we use a comprehensive search strategy to find new binary linear codes in the well-known and intensively studied class of quasi-cyclic (QC) co...

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R ar...

Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, have a special place in algebraic coding theory. Among other things, many of the best-known or optimal codes have been obtained from these classes. In this work we introduce a new generalization of QT codes that we call multi-twisted (MT) codes and study some of their...

We first define a new Gray map from R=Z4+uZ4 to Z4², where u²=1 and study (1+2u)-constacyclic codes over R. Also of interest are some properties of (1+2u)-constacyclic codes over R. Considering their Z4 images, we prove that the Gray images of (1+2u)-constacyclic codes of length n over R are cyclic codes of length 2n over Z4. In many cases the latt...

In this work, cyclic isodual codes over finite chain rings are investigated. These codes are monomially equivalent to their duals. Existence results for cyclic isodual codes are given based on the generator polynomials, the field characteristic, and the length. Several constructions of isodual and self-dual codes are also presented.

In this paper, we study constacyclic codes over finite principal ideal rings. An isomorphism between constacyclic codes and cyclic codes over finite principal ideal rings is given. Further, an open question is partially answered by giving necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite princip...

One of the most important and challenging problems in coding theory is to construct codes with good parameters. There are various methods to construct codes with the best possible parameters. A promising and fruitful approach has been to focus on the class of quasi-twisted (QT) codes which includes constacyclic codes as a special case. This class o...

Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, have a special place in algebraic coding theory. Among other things, many of the best-known or optimal codes have been obtained from these classes. In this work we introduce a new generalization of QT codes that we call multi-twisted (MT) codes and study some of their...

Abstract: In this paper, we study cyclic codes over the ring R = Z4+uZ4+u2Z4 , where u3 = 0. We investigate Galois extensions of this ring and the ideal structure of these extensions. The results are then used to obtain facts about cyclic codes over R. We also determine the general form of the generator of a cyclic code and �nd its minimal spanning...

Motivated by a generalization of the football pool problem, we introduce additive cyclic codes over mixed alphabets of the form 픽q1 × 픽q2 where q1 = p1m1, q2 = p2m2 for distinct primes p1,p2. We study their algebraic properties, generating sets, and duals. Additionally, we give examples of additive cyclic codes over the alphabet ℤ2ℤ3 that have best...

Additive codes received much attention due to their connections with quantum codes. On the other hand skew cyclic codes proved to be a useful class of codes that contain many good codes. In this work, we introduce and study additive skew cyclic codes over the quater-nary field GF (4), obtaining some structural properties of these codes. Moreover, w...

http://dx.doi.org/10.1016/j.ffa.2015.12.003

In this paper, we study negacyclic codes of length 2k over the ring \(R=\mathbb {Z}_{4}+u\mathbb {Z}_{4}\), u 2 = 0. We have obtained a mass formula for the number of negacyclic of length 2k over R. We have also determined the number of self-dual negacyclic codes of length 2k over R. This study has been further generalized to negacyclic codes of an...

There has been much research on codes over $\mathbb{Z}_4$, sometimes called quaternary codes, for over a decade. Yet, no database is available for best known quaternary codes. This work introduces a new database for quaternary codes. It also presents a new search algorithm called genetic code search (GCS), as well as new quaternary codes obtained b...

In this paper, we study the extraction of roots as presented by Al-Kashi in his 1427 book “Key to Arithmetic” and Stevin in his 1585 book “Arithmetic”. In analyzing their methods, we note that Stevin’s technique contains some flaws that we amend to present a coherent algorithm. We then show that the underlying algorithm for the methods of both Al-K...

One fundamental and challenging problem in coding theory is to optimize the parameters [n, k, d] of a linear code over the finite field Fq and construct codes with best possible parameters. There are tables and databases of best-known linear codes over the finite fields of size up to 9 together with upper bounds on the minimum distances. Motivated...

Error control codes have been widely used in data communications and storage systems. One central problem in coding theory is to optimize the parameters of a linear code and construct codes with best possible parameters. There are tables of best-known linear codes over finite fields of sizes up to 9. Recently, there has been a growing interest in c...

A classification of all four-circulant extremal codes of length 32 over F-2 + uF(2) is done by using four-circulant binary self-dual codes of length 32 of minimum weights 6 and 8. As Gray images of these codes, a substantial number of extremal binary self-dual codes of length 64 are obtained. In particular a new code with beta = 80 in W-64,W-2 is f...

One of the main challenges of coding theory is to construct linear codes with the best possible parameters. Various algebraic and combinatorial methods along with computer searches are used to construct codes with better parameters. Given the computational complexity of determining the minimum distance of a code, exhaustive searches are not feasibl...

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...

We consider quasi-cyclic codes over the ring F2 + u F 2 + v F2 + u v F2, a finite non-chain ring that has been recently studied in coding theory. The Gray images of these codes are shown to be binary quasi-cyclic codes. Using this method we have obtained seventeen new binary quasi-cyclic codes that are new additions to the database of binary quasi-...

In this paper we study θ-cyclic codes over the ring R = F 2 + vF 2 = {0, 1, v, v +1} where v 2 = v. This is the only ring of order four that is not a field and has a non-trivial ring automorphism. We describe generator polynomials of θ-cyclic codes defined over this ring. We also describe the generator polynomials of the duals of free θ-cyclic code...

In this paper we study θ-cyclic codes over the ring R = F 2 + υF 2 = {0, 1, υ, υ+1} where υ 2 = υ. This is the only ring of order four that is not a field and has a non-trivial ring automorphism. We describe generator polynomials of θ-cyclic codes defined over this ring. We also describe the generator polynomials of the duals of free θ-cyclic codes...

Students who are actively engaged in learning mathematics understand the subject better. In fact, one of the main problems of mathematics education today is to find ways to motivate and engage students in the classroom. In this quantitative study, cryptographic activities are used as an aid in teaching the topic of modular arithmetic in 8th grade a...

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...

One of the central problems in algebraic coding theory is construction of linear codes with best possible parameters. Quasi-twisted (QT) codes have been promising to solve this problem. Despite extensive search in this class and discovery of a large number of new codes, we have been able to find still more new codes that are QT over the alphabet F5...

For over a decade, there has been considerable research on codes over ℤ 4 and other rings. In spite of this, no tables or databases exist for codes over ℤ 4 , as it is the case with codes over finite fields. The purpose of this work is to contribute to the creation of such a database. We consider cyclic, negacyclic and quasi-twisted (QT) codes over...

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...

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) )...

This chapter gives an introduction to algebraic coding theory and a survey of constructions of some of the well-known classes of algebraic block codes such as cyclic codes, BCH codes, Reed-Solomon codes, Hamming codes, quadratic residue codes, and quasi-cyclic (QC) codes. It then describes some recent generalizations of QC codes and open problems r...

The theory of error-correcting codes and cryptography are two relatively recent applications of mathematics to information and communication systems. The mathematical tools used in these fields generally come from algebra, elementary number theory, and combinatorics, including concepts from computational complexity. It is possible to introduce the...

The purpose of this study is to investigate possible effects of different college level mathematics courses on college students' van Hiele levels of geometric understanding. Particularly, since logical reasoning is an important aspect of geometric understanding, it would be interesting to see whether there are differences in van Hiele levels of stu...