Quantum codes from cyclic codes over F3 + vF3

2014 ◽  
Vol 12 (06) ◽  
pp. 1450042 ◽  
Author(s):  
Mohammad Ashraf ◽  
Ghulam Mohammad
Keyword(s):  
Commutative Ring ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Quantum Codes ◽  
Quantum Code ◽  
Chain Ring ◽  
Arbitrary Length ◽  
Orthogonal Codes ◽  
Finite Commutative Ring ◽  
A Chain

Let R = F3 + vF3 be a finite commutative ring, where v2 = 1. It is a finite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over R. We derive self-orthogonal codes over F3 as Gray images of linear and cyclic codes over R. In particular, we use two codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code.

On quantum codes obtained from cyclic codes over 𝔽2 + u𝔽2 + u2𝔽2

2017 ◽  
Vol 11 (01) ◽  
pp. 1850009 ◽  
Author(s):  
Abhay Kumar Singh ◽  
Sukhamoy Pattanayak ◽  
Pratyush Kumar ◽  
Kar Ping Shum
Keyword(s):  
Commutative Ring ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Quantum Codes ◽  
Orthogonal Codes

In the paper, we consider a finite non-chain commutative ring [Formula: see text], where [Formula: see text]. We mainly study the construction of quantum codes from cyclic codes over [Formula: see text]. For this, we obtained self-orthogonal codes over [Formula: see text] as gray images of linear and cyclic codes over [Formula: see text], then these codes over [Formula: see text] correspond to a cyclic code over [Formula: see text] of odd length [Formula: see text] used to determine the parameters of the quantum codes.

QUATERNARY CONSTRUCTION OF QUANTUM CODES FROM CYCLIC CODES OVER $\mathbb{F}_4 + u\mathbb{F}_4$

2011 ◽  
Vol 09 (02) ◽  
pp. 689-700 ◽  
Author(s):  
XIAOSHAN KAI ◽  
SHIXIN ZHU
Keyword(s):  
Cyclic Code ◽  
Cyclic Codes ◽  
Binary Codes ◽  
Quantum Codes ◽  
Quantum Code ◽  
Orthogonal Codes

We give a construction for quantum codes from linear and cyclic codes over [Formula: see text]. We derive Hermitian self-orthogonal codes over [Formula: see text] as Gray images of linear and cyclic codes over [Formula: see text]. In particular, we use two binary codes associated with a cyclic code over [Formula: see text] of odd length to determine the parameters of the corresponding quantum code.

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.

ON TWO PROBLEMS OF ASYMMETRIC QUANTUM CODES

2014 ◽  
Vol 28 (06) ◽  
pp. 1450017 ◽  
Author(s):  
RUIHU LI ◽  
GEN XU ◽  
LUOBIN GUO
Keyword(s):  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Bch Code ◽  
Quantum Codes ◽  
Quantum Code ◽  
Asymmetric Quantum ◽  
Asymmetric Quantum Codes ◽  
Special Cases ◽  
Quantum Error ◽  
Better Than

In this paper, we discuss two problems on asymmetric quantum error-correcting codes (AQECCs). The first one is on the construction of a [[12, 1, 5/3]]2 asymmetric quantum code, we show an impure [[12, 1, 5/3 ]]2 exists. The second one is on the construction of AQECCs from binary cyclic codes, we construct many families of new asymmetric quantum codes with dz> δ max +1 from binary primitive cyclic codes of length n = 2m-1, where δ max = 2⌈m/2⌉-1 is the maximal designed distance of dual containing narrow sense BCH code of length n = 2m-1. A number of known codes are special cases of the codes given here. Some of these AQECCs have parameters better than the ones available in the literature.

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.

On quantum codes obtained from cyclic codes over A2

2015 ◽  
Vol 13 (03) ◽  
pp. 1550031 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Gray Map ◽  
Quantum Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Arbitrary Length ◽  
Necessary And Sufficient ◽  
Quantum Error

In this paper, quantum codes from cyclic codes over A2 = F2 + uF2 + vF2 + uvF2, u2 = u, v2 = v, uv = vu, for arbitrary length n have been constructed. It is shown that if C is self orthogonal over A2, then so is Ψ(C), where Ψ is a Gray map. A necessary and sufficient condition for cyclic codes over A2 that contains its dual has also been given. Finally, the parameters of quantum error correcting codes are obtained from cyclic codes over A2.

Quantum codes over a class of finite chain ring

2016 ◽  
pp. 39-49
Author(s):  
Mustafa Sari ◽  
Irfan Siap
Keyword(s):  
Finite Field ◽  
Cyclic Codes ◽  
Gray Map ◽  
Quantum Codes ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Quantum Error

In this study, we introduce a new Gray map which preserves the orthogonality from the chain ring F_2 [u] / (u^s ) to F^s_2 where F_2 is the finite field with two elements. We also give a condition of the existence for cyclic codes of odd length containing its dual over the ring F_2 [u] / (u^s ) . By taking advantage of this Gray map and the structure of the ring, we obtain two classes of binary quantum error correcting (QEC) codes and we finally illustrate our results by presenting some examples with good parameters.

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.

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