Cyclic codes over $${\mathbb {F}}_2 +u{\mathbb {F}}_2+v{\mathbb {F}}_2 +v^2 {\mathbb {F}}_2 $$F2+uF2+vF2+v2F2 with respect to the homogeneous weight and their applications to DNA codes

In this paper, we discuss the DNA construction of general length over the finite ring [Formula: see text], with [Formula: see text], which plays a very significant role in DNA computing. We discuss the GC weight of DNA codes over [Formula: see text]. Several examples of reversible cyclic codes over [Formula: see text] are provided, whose [Formula: see text]-images are [Formula: see text]-linear codes with good parameters.

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Skew cyclic codes over 𝔽4R

Journal of Algebra and Its Applications ◽

10.1142/s0219498822500657 ◽

2020 ◽

pp. 2250065

Author(s):

Nasreddine Benbelkacem ◽

Martianus Frederic Ezerman ◽

Taher Abualrub ◽

Nuh Aydin ◽

Aicha Batoul

Keyword(s):

Error Control ◽

Cyclic Codes ◽

Gray Map ◽

Inner Product ◽

Algebraic Structures ◽

Error Control Codes ◽

Dna Codes ◽

Natural Connection ◽

Skew Cyclic Codes

This paper considers a new alphabet set, which is a ring that we call [Formula: see text], 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 [Formula: see text]-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over [Formula: see text] are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of [Formula: see text]-skew cyclic codes which are reversible complement.

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New DNA Codes from Cyclic Codes over Mixed Alphabets

Mathematics ◽

10.3390/math8111977 ◽

2020 ◽

Vol 8 (11) ◽

pp. 1977

Author(s):

Hai Q. Dinh ◽

Sachin Pathak ◽

Ashish Kumar Upadhyay ◽

Woraphon Yamaka

Keyword(s):

Cyclic Codes ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Block Length ◽

Dna Codes ◽

Reverse Complement ◽

Gray Maps ◽

Cyclic Dna Codes ◽

One To One ◽

Necessary And Sufficient

Let R=F4+uF4,withu2=u and S=F4+uF4+vF4,withu2=u,v2=v,uv=vu=0. In this paper, we study F4RS-cyclic codes of block length (α,β,γ) and construct cyclic DNA codes from them. F4RS-cyclic codes can be viewed as S[x]-submodules of Fq[x]⟨xα−1⟩×R[x]⟨xβ−1⟩×S[x]⟨xγ−1⟩. We discuss their generator polynomials as well as the structure of separable codes. Using the structure of separable codes, we study cyclic DNA codes. By using Gray maps ψ1 from R to F42 and ψ2 from S to F43, we give a one-to-one correspondence between DNA codons of the alphabets {A,T,G,C}2,{A,T,G,C}3 and the elements of R,S, respectively. Then we discuss necessary and sufficient conditions of cyclic codes over F4, R, S and F4RS to be reversible and reverse-complement. As applications, we provide examples of new cyclic DNA codes constructed by our results.

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On cyclic DNA codes over the rings Z4 + wZ4 and Z4 + wZ4 + vZ4 + wvZ4

BIOMATH ◽

10.11145/j.biomath.2017.12.167 ◽

2017 ◽

Vol 6 (2) ◽

pp. 1712167 ◽

Cited By ~ 2

Author(s):

Abdullah Dertli ◽

Yasemin Cengellenmis

Keyword(s):

Cyclic Codes ◽

Finite Ring ◽

Finite Rings ◽

Binary Images ◽

Dna Codes ◽

Reverse Complement ◽

Skew Cyclic Codes ◽

Cyclic Dna Codes

The structures of cyclic DNA codes of odd length over the finite rings R = Z4 + wZ4, w^2 = 2 and S = Z4 + wZ4 + vZ4 + wvZ4; w^2 = 2; v^2 =v; wv = vw are studied. The links between the elements of the rings R, S and 16 and 256 codons are established, respectively. The cyclic codes of odd length over the finite ring R satisfy reverse complement constraint and the cyclic codes of odd length over the finite ring S satisfy reverse constraint and reverse complement constraint are studied. The binary images of the cyclic DNA codes over the finite rings R and S are determined. Moreover, a family of DNA skew cyclic codes over R is constructed, its property of being reverse complement is studied.

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The homogeneous weight for Rk, related gray map and new binary quasi-cyclic codes

Filomat ◽

10.2298/fil1704885y ◽

2017 ◽

Vol 31 (4) ◽

pp. 885-897 ◽

Cited By ~ 1

Author(s):

Bahattin Yildiz ◽

Ismail Kelebek

Keyword(s):

Coding Theory ◽

Cyclic Codes ◽

Gray Map ◽

Binary Codes ◽

Binary Images ◽

Frobenius Rings ◽

Homogeneous Weights ◽

Homogeneous Weight ◽

Theoretical Results ◽

Quasi Cyclic Codes

Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family Rk, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We find an associated Gray map for the homogeneous weight using first order Reed-Muller codes and we describe some of the general properties of the images of codes over Rk under this Gray map. We then discuss quasi-twisted codes over Rk and their binary images under the homogeneous Gray map. In this way, we find many optimal binary codes which are self-orthogonal and quasi-cyclic. In particular, we find a substantial number of optimal binary codes that are quasi-cyclic of index 8, 16 and 24, nearly all of which are new additions to the database of quasi-cyclic codes kept by Chen.

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The structure of duals of cyclic codes over 𝔽2 + u𝔽2 + v𝔽2 + uv𝔽2 and some DNA codes

International Journal of Information and Coding Theory ◽

10.1504/ijicot.2017.10002267 ◽

2017 ◽

Vol 4 (1) ◽

pp. 79

Author(s):

Maheshanand Bhaintwal ◽

Srinivasulu Bathala

Keyword(s):

Cyclic Codes ◽

Dna Codes

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DNA cyclic codes over the ring 𝔽2[u,v]/〈u2 − 1,v3 − v,uv − vu〉

International Journal of Biomathematics ◽

10.1142/s1793524518500420 ◽

2018 ◽

Vol 11 (03) ◽

pp. 1850042 ◽

Cited By ~ 1

Author(s):

Hai Q. Dinh ◽

Abhay Kumar Singh ◽

Sukhamoy Pattanayak ◽

Songsak Sriboonchitta

Keyword(s):

Cyclic Code ◽

Cyclic Codes ◽

Sufficient Conditions ◽

Gray Map ◽

Arbitrary Length ◽

Binary Images ◽

Dna Codes ◽

Cyclic Dna Codes ◽

Living Organisms ◽

Necessary And Sufficient

In this paper, our main objective is to find out the necessary and sufficient conditions for a cyclic code of arbitrary length over the ring of four elements [Formula: see text] [Formula: see text] to be a reversible cyclic code. We also obtain the structure of cyclic DNA codes of odd length over the ring [Formula: see text], which plays an important role in Computational Biology. Furthermore, we establish a direct link between the elements of ring [Formula: see text] and 64 codons used in the amino acids of living organisms by introducing a Gray map from [Formula: see text] to [Formula: see text]. Among others, binary images of cyclic codes over [Formula: see text] are also investigated. As applications, some cyclic DNA codes over [Formula: see text] using the Gray map are provided.

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