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Öğe APPLICATION OF CONSTACYCLIC CODES OVER THE SEMI LOCAL RING Fpm + vFpm(Indian Nat Sci Acad, 2020) Bag, Tushar; Dertli, Abdullah; Cengellenmis, Yasemin; Upadhyay, Ashish K.In this paper, we study the quantum codes over , which are obtained from (lambda(1) + lambda(2))-constacyclic codes over the semi local ring , where v(2) = 1, p is odd prime. We decompose a (lambda(1) + lambda(2))-constacyclic code over into two constacyclic codes over such as (lambda(1) + lambda(2))-constacyclic and (lambda(1)-lambda(2))-constacyclic. We give the necessary and sufficient condition that the (lambda(1) + v lambda(2))-constacyclic codes over contain their duals. We give some examples of non binary quantum codes.Öğe Codes over an infinite family of rings with a Gray map(Springer, 2014) Cengellenmis, Yasemin; Dertli, Abdullah; Dougherty, S. T.Codes over an infinite family of rings which are an extension of the binary field are defined. Two Gray maps to the binary field are attached and are shown to be conjugate. Euclidean and Hermitian self-dual codes are related to binary self-dual and formally self-dual codes, giving a construction of formally self-dual codes from a collection of arbitrary binary codes. We relate codes over these rings to complex lattices. A Singleton bound is proved for these codes with respect to the Lee weight. The structure of cyclic codes and their Gray image is studied. Infinite families of self-dual and formally self-dual quasi-cyclic codes are constructed from these codes.Öğe CYCLIC AND SKEW CYCLIC CODES OVER THE Fq(Editura Bibliotheca-Bibliotheca Publ House, 2016) Dertli, Abdullah; Cengellenmis, YaseminIn this paper, we study the structure of cyclic and skew cyclic codes over the finite ring D-k = F-q +v(1)F(q) + ... v(k)F(q), v(i)(2) = v(i), v(i)v(j) = v(j)v(i) = 0, 1 <= i, j <= k, q = p(m), p is a prime for k >= 1 which contains the ring F-q + v(1)F(q), v(1)(2) = v(1). We define a new Gray map from D-k to F-q(k+1). The algebraic structures of cyclic codes and duality properties are investigated. A linear code over D-k is represented by means of k+1 q-ary codes. The non trivial automorphism over D-k is given and the skew cyclic codes over D-k are introduced. The algebraic structure of skew cyclic codes and duality properties are investigated. The Gray images of both cyclic and skew cyclic codes over D-k are obtained.Öğe The gray image of the cyclic codes over IF2[u]/?u2 - 1?(Taru Publications, 2009) Cengellenmis, YaseminIt is proved that every binary code which is the Gray image of a linear cyclic code over IF2[u]/< u(2)-1 > with length n is equivalent to a linear cyclic code, if n is odd.Öğe A new matrix form to generate all 3 x 3 involutory MDS matrices over F2m(Elsevier Science Bv, 2019) Guzel, Gulsum Gozde; Sakalli, Muharrem Tolga; Akleylek, Sedat; Rijmen, Vincent; Cengellenmis, YaseminIn this paper, we propose a new matrix form to generate all 3 x 3 involutory and MDS matrices over F-2(m) and prove that the number of all 3 x 3 involutory and MDS matrices over F-2(m) is (2(m) - 1)(2) . (2(m) - 2) . (2(m) - 4), where m > 2. Moreover, we give 3 x 3 involutory and MDS matrices over F-2(3), F-2(4) and F-2(8) defined by the irreducible polynomials x(3) +x+ 1, x(4) +x + 1 and x(8) + x(7) + x(6) + x + 1, respectively, by considering the minimum XOR count, which is a metric used in the estimation of hardware implementation cost. Finally, we provide the maximum number of 1s in 3 x 3 involutory MDS matrices. (C) 2019 Elsevier B.V. All rights reserved.Öğe On (1(European Journal Pure & Applied Mathematics, 2016) Dertli, Abdullah; Cengellenmis, YaseminIt is studied codes over the ring R = F-2 + uF(2) + vF(2) , u(2) = 0, v(2) = v, vu = vu = 0 which contains the two ring F-2 + uF(2), u(2) = 0 and F-2 +vF(2), v(2) = v. It is introduced (1 + u) -cyclic codes and cyclic codes over F-2 + uF(2) + vF(2). It is characterized codes over F-2 + vF(2) which are the images of (1 + u)-cyclic codes and cyclic codes over F-2 + uF(2) + vF(2). It is obtained a representation of a linear code of length n over R by means of C-1 and C-2 which are linear codes of length n over F-2 + uF(2). It is also characterized codes over F-2 which are the Gray images of (1 + u)-cyclic codes or cyclic codes over F-2 + uF(2) + vF(2).Öğe On (1 - u2)-cyclic and cyclic codes over IFp(Taru Publications, 2009) Cengellenmis, YaseminIt is extended a result of [5] to codes over the commutative ring IFp + uIF(p) + u(2)IFp where u(3) = 0. A new Gray map between codes over IFp + uIF(p) + u(2)IFp and IFp is defined. It is proved that the Gray image of the linear (1 - u(2))-cyclic code over the commutative ring IFp + uIF(p) + u(2)IF(p) of length n is a distance-invariant quasi-cyclic code of index p and length p(2)n over IFp. And it is proved that if (n, p) = 1, then every code of length p(2)n over IFp which is the Gray image of a linear cyclic code of length n over IFp + uIF(p) + u(2)IF(p) is permutation equivalent to a quasi-cyclic code of index p.Öğe ON GRAY IMAGES OF CONSTACYCLIC CODES OVER THE FINITE RING F2 + u1F2 + u2F2(Turkic World Mathematical Soc, 2019) Ozkan, Mustafa; Dertli, Abdullah; Cengellenmis, YaseminWe introduce the finite ring F-2 + u(1)F(2) + u(2)F(2), u(1)(2) = u(1), u(2)(2) = 0, u(1).u(2) = u(2).u(1) = 0 which is not a finite chain ring. We focus on (1 + u(2)) -constacyclic codes over F-2 + u(1)F(2) + u(2)F(2) of odd length. We prove that the Gray image of a linear (1 + u(2)) -constacyclic code over F-2 + u(1)F(2) + u(2)F(2) of odd length n is a quasi-cyclic code of index 4 and length 4 n over F-2.Öğe On quantum codes obtained from cyclic codes over A2(World Scientific Publ Co Pte Ltd, 2015) Dertli, Abdullah; Cengellenmis, Yasemin; Eren, SenolIn this paper, quantum codes from cyclic codes over A(2) = F-2 + uF(2) + vF(2) + uvF(2), u(2) = u, v(2) = v, uv = vu, for arbitrary length n have been constructed. It is shown that if C is self orthogonal over A(2), then so is Psi(C), where Psi is a Gray map. A necessary and sufficient condition for cyclic codes over A(2) that contains its dual has also been given. Finally, the parameters of quantum error correcting codes are obtained from cyclic codes over A(2).Öğe On skew cyclic codes over a mixed alphabet and their applications to DNA codes(World Scientific Publ Co Pte Ltd, 2022) Dertli, Abdullah; Cengellenmis, Yasemin; Aydin, NuhIn this paper, we introduce skew cyclic codes over the mixed alphabet F4R, where F-4 is the finite field with 4 elements and R = F-4 + uF(4) + vF(4), u(2) = u , v(2) = v , uv = vu = 0. Our results include a description of the generator polynomials of such codes and a necessary and sufficient condition for an F4R-skew cyclic code to be reversible complement.Öğe On some constacyclic codes over , their images, and new codes(Springer, 2018) Aydin, Nuh; Cengellenmis, Yasemin; Dertli, AbdullahIn this paper, we study -constacyclic codes over the ring where , for and . Two new Gray maps from R to are defined with the goal of obtaining new linear codes over . The Gray images of -constacyclic codes over R are determined. We then conducted a computer search and obtained many -constacyclic codes over R whose -images have better parameters than currently best-known linear codes over Z(4).Öğe On the Codes over a Semilocal Finite Ring(Science & Information Sai Organization Ltd, 2015) Dertli, Abdullah; Cengellenmis, Yasemin; Eren, SenolIn this paper, we study the structure of cyclic, quasi cyclic, constacyclic codes and their skew codes over the finite ring R. The Gray images of cyclic, quasi cyclic, skew cyclic, skew quasi cyclic and skew constacyclic codes over R are obtained. A necessary and sufficient condition for cyclic (negacyclic) codes over R that contains its dual has been given. The parameters of quantum error correcting codes are obtained from both cyclic and negacyclic codes over R. Some examples are given. Firstly, quasi constacyclic and skew quasi constacyclic codes are introduced. By giving two inner product, it is investigated their duality. A sufficient condition for 1 generator skew quasi constacyclic codes to be free is determined.Öğe On the linear codes over the ring Rp(World Scientific Publ Co Pte Ltd, 2016) Dertli, Abdullah; Cengellenmis, Yasemin; Eren, SenolSome results are generalized on linear codes over Z(3)[v]/< v(3) - v > in [15] to the ring R-p = Z(p)[v]/< v(p) - v >, where p is an odd prime number. The Gray images of cyclic and quasi-cyclic codes over R-p are obtained. The parameters of quantum error correcting codes are obtained from negacyclic codes over R-p. A nontrivial automorphism theta(p) on the ring R-p is determined. By using this, the skew cyclic, skew quasi-cyclic, skew constacyclic codes over R-p are introduced. The number of distinct skew cyclic codes over R-p is given. The Gray images of skew codes over R-p are obtained. The quasi-constacyclic and skew quasi-constacyclic codes over R-p are introduced. MacWilliams identities of linear codes over R-p are given.Öğe ON THE LINEAR CODES OVER THE RING Z4(Ankara Univ, Fac Sci, 2019) Dertli, Abdullah; Cengellenmis, YaseminSome results on linear codes over the ring Z(4) + uZ(4) + vZ(4); u(2) = u, v(2) = v; uv = vu = 0 in [6,7] are generalized to the ring D-t = Z(4) + v(1)Z(4) + ... + v(t)Z(4); v(i)(2) = v(i),v(i)v(j) = v(j)v(i) = 0 for i not equal j; 1 <= i, j <= t. A Gray map Phi(t) from D-t(n) to Z(4)((t+1)n) is defined. The Gray images of the cyclic, constacyclic and quasi-cyclic codes over Dt are determined. The cyclic DNA codes over D-t are introduced. The binary images of them are determined. The nontrivial automorphism on D-i for i = 2; 3; ..., t is given. The skew cyclic, skew constacyclic and skew quasi-cyclic codes over D-t are introduced. The Gray images of them are determined. The skew cyclic DNA codes over D-t are introduced. Moreover, some properties of MDS codes over D-t are discussed.Öğe ON THE QUANTUM CODES OVER Yq(Editura Bibliotheca-Bibliotheca Publ House, 2020) Dertli, Abdullah; Cengellenmis, YaseminIn this paper, the quantum codes over F-q are constructed by using the cyclic codes over with, and is an Y-q = F-q + uF(q) + uvF(q) with u(2) = 1, v(2) = 1, uv = vu, and q = p(m), p is an odd prime. Moreover, the parameters of quantum codes over F-q are determined.Öğe On the Skew Cyclic Codes and the Reversibility Problem for DNA 4-Bases(Springer Basel Ag, 2020) Cengellenmis, Yasemin; Dertli, AbdullahThe DNA 4-bases are matched with the 256 elements of the finite ring R=F4+uF4+vF4+uvFR=F_{4}+uF_{4}+vF_{4}+uvF_{4}$$\end{document}, where u2=u,v2=v,uv=vu. By defining a non trivial automorphism over R, the skew cyclic codes over the finite ring R are introduced. The reversible DNA codes are obtained, by using the skew cyclic codes whose generator polynomials satisfy some conditions. Thanks to this, the reversibility problem for DNA 4-bases can be solved. Moreover, the Gray images of the skew cyclic codes over the finite ring are determined.Öğe Quantum Codes Obtained from Some Constacyclic Codes Over a Family of Finite Rings Fp+uFp+vFp(Springer Basel Ag, 2020) Dertli, Abdullah; Cengellenmis, YaseminIn this paper, we study the quantum codes over Fp, p is an odd prime, obtained from some constacyclic codes over the finite ring Fp+uFp+vFp, where u2=u,v2=v,uv=vu=0. A constacyclic code over the finite ring Fp+uFp+vFp is decomposed into three codes over Fpin order to determine the parameters of the corresponding quantum codes. Finally, we have constructed some examples of quantum error-correcting codes.Öğe REVERSIBLE G-CODES OVER THE RING Fj,k WITH APPLICATIONS TO DNA CODES(Amer Inst Mathematical Sciences-Aims, 2023) Cengellenmis, Yasemin; Dertli, Abdullah; Dougherty, Steven T.; Korban, Adrian; Sahinkaya, Serap; Ustun, DenizIn this paper, we show that one can construct a G-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which G-codes are reversible of index alpha. Additionally, we introduce a new family of rings, F-j,F-k, whose base is the finite field of order 4 and study reversible G-codes over this family of rings. Moreover, we present some possible applications of reversible G-codes over F-j,F-k to reversible DNA codes. We construct many reversible G-codes over F-4 of which some are optimal. These codes can be used to obtain reversible DNA codes.Öğe SKEW CONSTACYCLIC CODES OVER THE LOCAL FROBENIUS NON-CHAIN RINGS OF ORDER 16(Amer Inst Mathematical Sciences-Aims, 2020) Aydin, Nuh; Cengellenmis, Yasemin; Dertli, Abdullah; Dougherty, Steven T.; Salturk, EsengulWe introduce 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 some of these rings.Öğe SKEW CYCLIC CODES OVER Fq(Editura Bibliotheca-Bibliotheca Publ House, 2017) Dertli, Abdullah; Cengellenmis, YaseminIn this paper, the skew cyclic codes over the ring R = F-q + uF(q) + vF(q) + uvF(q), where u(2) = 1, v(2) = 1, uv = vu, q = pm, p is an odd prime are studied. The structural properties of them are investigated.