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.

scholarly journals New quantum codes from skew constacyclic codes

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ram Krishna Verma ◽  
Om Prakash ◽  
Ashutosh Singh ◽  
Habibul Islam
Keyword(s):  
Finite Field ◽  
Gray Map ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Dual Codes ◽  
Chain Ring ◽  
Css Construction ◽  
Positive Integers

<p style='text-indent:20px;'>For an odd prime <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula> and positive integers <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \ell $\end{document}</tex-math></inline-formula>, let <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{F}_{p^m} $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M5">\begin{document}$ p^{m} $\end{document}</tex-math></inline-formula> elements and <inline-formula><tex-math id="M6">\begin{document}$ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $\end{document}</tex-math></inline-formula>. Thus <inline-formula><tex-math id="M7">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula> is a finite commutative non-chain ring of order <inline-formula><tex-math id="M8">\begin{document}$ p^{2^{\ell} m} $\end{document}</tex-math></inline-formula> with characteristic <inline-formula><tex-math id="M9">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, we aim to construct quantum codes from skew constacyclic codes over <inline-formula><tex-math id="M10">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula>. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.</p>

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.

scholarly journals New Quantum and LCD Codes over Finite Fields of Even Characteristic

2021 ◽  
Vol 71 (5) ◽  
pp. 656-661
Author(s):  
Habibul Islam ◽  
Om Prakash
Keyword(s):  
Finite Fields ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Gray Map ◽  
Chain Ring ◽  
Lcd Codes ◽  
Lcd Code ◽  
Quantum Error

For an integer m ≥ 1, we study cyclic codes of length l over a commutative non-chain ring F2m + uF2m , where u2 = u . With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length l with respect to their generator polynomials and prove that F2m − image of an LCD code of length l is an LCD code of length 2l . Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m .  

QUANTUM CODES FROM CYCLIC CODES OVER FINITE RING

2009 ◽  
Vol 07 (06) ◽  
pp. 1277-1283 ◽  
Author(s):  
JIANFA QIAN ◽  
WENPING MA ◽  
WANGMEI GUO
Keyword(s):  
Finite Field ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Finite Ring ◽  
New Method ◽  
Quantum Codes ◽  
Orthogonal Codes ◽  
Quantum Error

A new method to obtain self-orthogonal codes over finite field F2 is presented. Based on this method, we provide a construction for quantum error-correcting codes starting from cyclic codes over finite ring R = F2 + uF2. As an example, we present infinite families of quantum error-correcting codes which are derived from cyclic codes over the ring R = F2 + uF2.

Cyclic self-orthogonal codes over finite chain ring

2018 ◽  
Vol 11 (06) ◽  
pp. 1850078 ◽  
Author(s):  
Abhay Kumar Singh ◽  
Narendra Kumar ◽  
Kar Ping Shum
Keyword(s):  
Prime Number ◽  
Cyclic Codes ◽  
Sufficient Condition ◽  
Galois Rings ◽  
Finite Chain ◽  
Necessary And Sufficient Condition ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Orthogonal Codes ◽  
Necessary And Sufficient

In this paper, we study the cyclic self-orthogonal codes over a finite commutative chain ring [Formula: see text], where [Formula: see text] is a prime number. A generating polynomial of cyclic self-orthogonal codes over [Formula: see text] is obtained. We also provide a necessary and sufficient condition for the existence of nontrivial self-orthogonal codes over [Formula: see text]. Finally, we determine the number of the above codes with length [Formula: see text] over [Formula: see text] for any [Formula: see text]. The results are given by Zhe-Xian Wan on cyclic codes over Galois rings in [Z. Wan, Cyclic codes over Galois rings, Algebra Colloq. 6 (1999) 291–304] are extended and strengthened to cyclic self-orthogonal codes over [Formula: see text].

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