Glift codes over chain ring and non-chain ring Re,s

In this paper, Glift codes, generalized lifted polynomials, matrices are introduced. The advantage of Glift code is “distance preserving” over the ring R. Then optimal codes can be obtained over the rings by using Glift codes and lifted polynomials. Zero divisors are classified to satisfy “distance preserving” for codes over non-chain rings. Moreover, Glift codes apply on MDS codes and MDS codes are obtained over the ring R and the non-chain ring Re,s. © 2022 Korean Mathematical Society

DNA kodlarının cebirsel yapısı

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Üç bölüm halinde düzenlenen bu çalışmanın birinci bölümünde gerekli cebirsel tanımlar, teoremler ve lineer kodlarla ilgili bilgiler verilmektedir.İkinci bölümde DNA kodları hakkındaki temel tanım ve teoremler verildi. Ayrıca GF(4) üzerinde oluşturulmuş DNA kodları örneklendirildi.Üçüncü bölümde ise GF(16) da toplamsal sıralı terslenebilen, tamamlanan ve sıralı terslenebilen-tamamlanan devirli kodlar inşa edildi ve bazı örnekler verildi.This study consists of three chapters. First chapter includes algebraic definitions, theorems and some information for linear codes.In the second chapter, definition, theorems and examples in GF(4) about DNA codes are introduced.In the third chapter, additive reversible, complement and reversible complement cyclic codes in GF(16) are built and some examples are worked out

A novel method for determining the non-cds region by using error-correcting codes

Our main motivation question is "Is there any relation between the non-coding region and useless error-correcting codes?". Then we focused CDS and non-CDS areas instead of exon and intron, because CDS involves in process of synthesis a protein and is involved by exons. We get the data of the genes from NCBI [21]. In this study, we introduce the method Fi-noncds that is used for determining the non-CDS region by using error-correcting codes. We obtained that the error-correction codes that can't correct any codes named zero error-correcting code, placed in non-CDS areas, densely. This result shows that non-CDS regions (non-coding areas in DNA) match zero error-correcting codes (useless error-correcting code). Frame lengths 7,8,9 and 10,11,12,13 and 14 were tested by the method. Optimal result for selected genes (TRAV1-1, TRAV1-2, TRAV2, TRAV7, WRKY33, HY5, GR-RBP2) is frame length 8, n=7, k=2, dnaNo=1. Moreover, optimal results of the algorithm Fi-noncds matched the best sequence length 8 as in [1]

Cyclic codes over F-2 + uF(2) + vF(2) + v(2)F(2) with respect to the homogeneous weight and their applications to DNA codes

WOS:000515672100001In this paper, we study cyclic codes and their duals over the local Frobenius non-chain ring R = F-2[u, v]/, and we obtain optimal binary linear codes with respect to the homogeneous weight over R via a Gray map. Moreover, we characterize DNA codes as images of cyclic codes over R

M-adic residue codes over f-q[v]/(v(2) - v) and dna codes

WOS:000433164500019In this study we determine the structure of m-adic residue codes over the non-chain ring F-q[v]1 (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 codes via a special automorphism and sets over F-42k [v]/(v(2) - v)

Reversible DNA codes using skew polynomial rings

WOS:000406328600004In 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

Reversible DNA codes over F16 + uF16 + vF16 + uvF16

WOS:000401830400007In 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 more general k-DNA strings called k-mers have been matched with some special rings and codes over such rings with specific properties are studied. However, these matchings in general are not straight-forward and because of the fact that the reverse of the codewords (k-mers) need to exist in the code, the matching problem is difficult and it is referred to as the reversibility problem. Here, 8-mers (DNA 8-bases) are matched with the ring elements of R-16 = F-16 + uF(16) + upsilon F-16 + u upsilon F-16. Furthermore, cyclic codes over the ring R-16 where the multiplication is taken to be noncommutative with respect to an automorphism theta are studied. The preference on the skewness is shown to be very useful and practical especially since this serves as a direct solution to the reversibility problem compared to the commutative approaches

On perfect powers in k-generalized pell sequence

Let k 2 and let (P(k) n )n>2-k be the k-generalized Pell sequence defined byP(k) n = 2P(k) n-1 + P(kn-2) + ... + P(k) n-kfor n 2 with initial conditionsP(k) -(k-2)= P(k-(k) -3)= ... = P(k) -1 = P0(k) = 0, P(k) 1 = 1.In this study, we handle the equation Pn(k) = ym in positive integers n, m, y, k such that k, y 2, and give an upper bound on n. Also, we will show that the equation P(k) n = ym with 2 y 1000 has only one solution given by P7(2) = 132

Cyclic and constacyclic codes over the ring Z4u/(u3 – u2) and their Gray images

WOS:000651619000039In this article, the structure of generator polynomial of the cyclic codes with odd length is formed over the ring Z4 + uZ4 + u 2Z4 where u 3 = u 2 . With the isomorphism we have defined, the generator polynomial of constacyclic codes with odd length over this ring is created from the generator of the cyclic codes. Additionally, necessary and sufficient conditions for a linear code in this ring to be a self dual code and a LCD code are mentioned. Furthermore, for all units over this ring, Z4 -images of λ-constacyclic codes and also Z4 -images of cyclic codes are examined by using related ones from defined three new Gray maps. Moreover, several new and optimal codes are constructed in terms of the Lee, Euclidean and Hamming weight in reference to the database

Construction of quantum codes over the class of commutative rings and their applications to DNA codes

WOS:000961861500001Let (Formula presented.) be positive integers and (Formula presented.) be a finite field of order (Formula presented.) of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring (Formula presented.), for (Formula presented.), where (Formula presented.) is the non-zero element of (Formula presented.). As an application, we obtain better quantum error correcting codes over the ring (Formula presented.) (for (Formula presented.)). Moreover, we acquire optimal linear codes with the help of the Gray image of cyclic codes. Finally, we present methods for reversible DNA codes