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.

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 Constacyclic and Linear Complementary Dual Codes Over Fq + uFq

2020 ◽  
Vol 70 (6) ◽  
pp. 626-632
Author(s):  
Om Prakash ◽  
Shikha Yadav ◽  
Ram Krishna Verma
Keyword(s):  
Cyclic Codes ◽  
Gray Map ◽  
Constacyclic Codes ◽  
Dual Codes ◽  
Gray Image ◽  
New Class ◽  
Sharing Scheme ◽  
Lcd Codes ◽  
Reversible Cyclic Codes

This article discusses linear complementary dual (LCD) codes over ℜ = Fq+uFq(u2=1) where q is a power of an odd prime p. Authors come up with a new Gray map from ℜn to F2nq and define a new class of codes obtained as the gray image of constacyclic codes over .ℜ Further, we extend the study over Euclidean and Hermitian LCD codes and establish a relation between reversible cyclic codes and Euclidean LCD cyclic codes over ℜ. Finally, an application of LCD codes in multisecret sharing scheme is given.

A Note on Skew Cyclic Codes over a Class of Rings

2020 ◽  
Vol 27 (04) ◽  
pp. 703-712
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta
Keyword(s):  
Finite Field ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
General Setting ◽  
Constacyclic Codes ◽  
Arbitrary Length ◽  
Skew Cyclic Codes

We study skew cyclic codes over a class of rings [Formula: see text], where each [Formula: see text] [Formula: see text] is a finite field. We prove that a skew cyclic code of arbitrary length over R is equivalent to either a usual cyclic code or a quasi-cyclic code over R. Moreover, we discuss possible extension of our results in the more general setting of [Formula: see text]-dual skew constacyclic codes over R, where δR is an automorphism of R.

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.

A note on cyclic codes over ℤ4 + uℤ4

2016 ◽  
Vol 08 (01) ◽  
pp. 1650017 ◽  
Author(s):  
Rama Krishna Bandi ◽  
Maheshanand Bhaintwal
Keyword(s):  
Local Ring ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Ideal Structure ◽  
Complete Ideal ◽  
Spanning Set

In this paper, we have studied cyclic codes over the ring [Formula: see text], [Formula: see text]. We have provided the general form of the generators of a cyclic code over [Formula: see text] and obtained a minimal spanning set for such codes and determined their ranks. We have determined a necessary condition and a sufficient condition for cyclic codes over [Formula: see text] to be [Formula: see text]-free. For [Formula: see text], we have shown that [Formula: see text] is a local ring, and the complete ideal structure of [Formula: see text] is determined. Some examples are presented.

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.

On quantum codes via cyclic codes of arbitrary length over 𝔽4 + u𝔽4

2018 ◽  
Vol 10 (03) ◽  
pp. 1850033 ◽  
Author(s):  
Amit Sharma ◽  
Ramakrishna Bandi ◽  
Maheshanand Bhaintwal
Keyword(s):  
Linear Code ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Quantum Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Arbitrary Length ◽  
Gray Image ◽  
Necessary And Sufficient

In this paper, we study cyclic codes over [Formula: see text]. A necessary and sufficient condition for a cyclic code over [Formula: see text] to contain its dual is determined. The odd and even length cases are discussed separately to obtain above condition. It is shown that Gray image of a cyclic code over [Formula: see text] containing its dual is a linear code over [Formula: see text] which also contains its dual. We have then obtained the parameters of corresponding CSS-quantum codes over [Formula: see text]. By augmentation, we construct codes with dual-containing property from codes of smaller size containing their duals. Through this construction, we have obtained some optimal quantum codes over [Formula: see text]. Some examples have been given to illustrate the results.

scholarly journals Ideal structure of Zq + uZq and Zq + uZq-cyclic codes

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4199-4214
Author(s):  
Raj Kumar ◽  
Maheshanand Bhaintwal ◽  
Ramakrishna Bandi
Keyword(s):  
Cyclic Code ◽  
Cyclic Codes ◽  
Gray Map ◽  
Sufficient Condition ◽  
Ideal Structure ◽  
Necessary And Sufficient Condition ◽  
Complete Ideal ◽  
Necessary And Sufficient

In this paper, we study cyclic codes of length n over R = Zq + uZq, u2 = 0, where q is a power of a prime p and (n; p) = 1. We have determined the complete ideal structure of R. Using this, we have obtained the structure of cyclic codes and that of their duals through the factorization of xn-1 over R. We have also computed total number of cyclic codes of length n over R. A necessary and sufficient condition for a cyclic code over R to be self-dual is presented. We have presented a formula for the total number of self-dual cyclic codes of length n over R. A new Gray map from R to Z2rp is defined. Using Magma, some good cyclic codes of length 4 over Z9 + uZ9 are obtained.

A note on skew constacyclic codes over 𝔽q + u𝔽q + v𝔽q

2019 ◽  
Vol 11 (03) ◽  
pp. 1950030
Author(s):  
Habibul Islam ◽  
Om Prakash
Keyword(s):  
Structural Properties ◽  
Decomposition Method ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Chain Ring ◽  
Gray Image

In this paper, the skew constacyclic codes over finite non-chain ring [Formula: see text], where [Formula: see text], [Formula: see text] is an odd prime and [Formula: see text], are studied. We show that the Gray image of a skew [Formula: see text]-constacyclic code of length [Formula: see text] over [Formula: see text] is a skew quasi-twisted code of length [Formula: see text] over [Formula: see text] of index 3. Further, the structural properties of skew constacyclic codes over [Formula: see text] are obtained by the decomposition method.

ℤ2(ℤ2 + uℤ2)-Additive cyclic codes and their duals

2016 ◽  
Vol 08 (02) ◽  
pp. 1650027 ◽  
Author(s):  
B Srinivasulu ◽  
Maheshanand Bhaintwal
Keyword(s):  
Structural Properties ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Free Module ◽  
Necessary Condition ◽  
Spanning Set

In this paper, we study some structural properties of [Formula: see text]-additive cyclic codes in [Formula: see text] as [Formula: see text]-submodules of [Formula: see text], where [Formula: see text]. The generators for these codes are obtained and a minimal spanning set is determined for even and odd [Formula: see text] separately with arbitrary [Formula: see text]. We also determine the generators of duals of the [Formula: see text]-additive cyclic codes for odd [Formula: see text]. A necessary condition for a 1-generator [Formula: see text]-cyclic code to be a [Formula: see text]-free module is obtained.

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