scholarly journals Skew cyclic codes over 𝔽4R

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.

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

BIOMATH ◽  
2017 ◽  
Vol 6 (2) ◽  
pp. 1712167 ◽  
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.

On skew cyclic codes over a semi-local ring

2015 ◽  
Vol 07 (04) ◽  
pp. 1550042 ◽  
Author(s):  
Mohammad Ashraf ◽  
Ghulam Mohammad
Keyword(s):  
Local Ring ◽  
Structural Properties ◽  
Decomposition Method ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Gray Map ◽  
Gray Image ◽  
Skew Cyclic Codes

In the present paper, we study skew cyclic codes over the finite semi-local ring [Formula: see text], where [Formula: see text] and [Formula: see text] is an odd prime. We define a Gray map from [Formula: see text] to [Formula: see text] and investigate the structural properties of skew cyclic codes over [Formula: see text] using decomposition method. It is proved that the Gray image of a skew cyclic code of length [Formula: see text] over [Formula: see text] is a skew [Formula: see text]-quasi-cyclic code of length [Formula: see text] over [Formula: see text]. Further, it is shown that the skew cyclic codes over [Formula: see text] are principally generated.

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

2018 ◽  
Vol 11 (03) ◽  
pp. 1850042 ◽  
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.

Skew cyclic codes over Fq + uFq + vFq

2018 ◽  
Vol 11 (05) ◽  
pp. 1850072 ◽  
Author(s):  
Mohammad Ashraf ◽  
Ghulam Mohammad
Keyword(s):  
Structural Properties ◽  
Decomposition Method ◽  
Cyclic Codes ◽  
Gray Map ◽  
Skew Cyclic Codes ◽  
Quasi Cyclic Codes

In this paper, we study skew cyclic codes over the ring [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is a prime. We define a Gray map from [Formula: see text] to [Formula: see text] and investigate the structural properties of skew cyclic codes over [Formula: see text] using decomposition method. It is shown that the Gray images of skew cyclic codes of length [Formula: see text] over [Formula: see text] are the skew [Formula: see text]-quasi cyclic codes of length [Formula: see text] over [Formula: see text]. Finally, the idempotent generators of skew cyclic codes over [Formula: see text] have also been discussed.

scholarly journals Construction of Dual Cyclic Codes over {F}_{2}[u,v]/ for DNA Computation

2018 ◽  
Vol 68 (5) ◽  
pp. 467-472
Author(s):  
Manoj Kumar Singh ◽  
Abhay Kumar Singh ◽  
Narendra Kumar ◽  
Pooja Mishra ◽  
Indivar Gupta
Keyword(s):  
Cyclic Codes ◽  
Gc Content ◽  
Inner Product ◽  
Dual Codes ◽  
Dna Computation ◽  
Dna Codes ◽  
Reverse Complement

Here, we assume the construction of cyclic codes over ℜ={F}_{2}[u,v]/ < u^2, v^2 - v, uv - vu >. In particular, dual cyclic codes over ℜ= {F}_{2}[u]/ <u^2> with respect to Euclidean inner product are discussed. The cyclic dual codes over ℜ are studied with respect to DNA codes (reverse and reverse complement). Many interesting results are obtained. Some examples are also provided, which explain the main results. The GC-Content and DNA codes over ℜ are discussed. We summarise the article by giving a special DNA table.

On skew cyclic codes over a mixed alphabet and their applications to DNA codes

2021 ◽  
pp. 2150143
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Nuh Aydin
Keyword(s):  
Finite Field ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dna Codes ◽  
Skew Cyclic Codes ◽  
Necessary And Sufficient

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 reversible complement.

scholarly journals On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 116
Author(s):  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
The Other ◽  
Inner Product ◽  
Equivalent Characterization ◽  
Geometric Constant ◽  
Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

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