scholarly journals Quantum Codes from Constacyclic Codes over the Ring Fq[u1,u2]/〈 u 1 2 -u1, u 2 2 -u2,u1u2-u2u1〉

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 781 ◽  
Author(s):  
Ahmad N. Alkenani ◽  
Mohammad Ashraf ◽  
Ghulam Mohammad
Keyword(s):  
Structural Properties ◽  
Linear Codes ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Gray Image

In this paper, we study the structural properties of ( α + u 1 β + u 2 γ + u 1 u 2 δ ) -constacyclic codes over R = F q [ u 1 , u 2 ] / ⟨ u 1 2 − u 1 , u 2 2 − u 2 , u 1 u 2 − u 2 u 1 ⟩ where q = p m for odd prime p and m ≥ 1 . We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4 n . Also we have classified all possible self dual linear codes over this ring R . We have given the applications by computing non-binary quantum codes over this ring R .

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.

Quantum codes from (1 − 2v)-constacyclic codes over the ring 𝔽q + u𝔽q + v𝔽q + uv𝔽q

2018 ◽  
Vol 10 (04) ◽  
pp. 1850046 ◽  
Author(s):  
Juan Li ◽  
Jian Gao ◽  
Yongkang Wang
Keyword(s):  
Structural Properties ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Chain Ring

In this paper, structural properties of [Formula: see text]-constacyclic codes over the finite non-chain ring [Formula: see text] are studied, where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is a power of some odd prime. As an application, some better quantum codes, compared with previous work, are obtained.

Quantum codes from a class of constacyclic codes over finite commutative rings

2019 ◽  
Vol 19 (12) ◽  
pp. 2150003 ◽  
Author(s):  
Hai Q. Dinh ◽  
Tushar Bag ◽  
Ashish K. Upadhyay ◽  
Mohammad Ashraf ◽  
Ghulam Mohammad ◽  
...  
Keyword(s):  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Negacyclic Codes ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Necessary And Sufficient

Let [Formula: see text] be an odd prime, and [Formula: see text] be an integer such that [Formula: see text]. Using pairwise orthogonal idempotents [Formula: see text] of the ring [Formula: see text], with [Formula: see text], [Formula: see text] is decomposed as [Formula: see text], which contains the ring [Formula: see text] as a subring. It is shown that, for [Formula: see text], [Formula: see text], and it is invertible if and only if [Formula: see text] and [Formula: see text] are units of [Formula: see text]. In such cases, we study [Formula: see text]-constacyclic codes over [Formula: see text]. We present a direct sum decomposition of [Formula: see text]-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a [Formula: see text]-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over [Formula: see text], with better parameters than existing ones, are constructed from cyclic and negacyclic codes over [Formula: see text].

scholarly journals On the construction of optimal asymmetric quantum codes

2014 ◽  
Vol 12 (03) ◽  
pp. 1450017 ◽  
Author(s):  
Liqi Wang ◽  
Shixin Zhu
Keyword(s):  
Linear Codes ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Asymmetric Quantum ◽  
Asymmetric Quantum Codes

Constacyclic codes are important classes of linear codes that have been applied to the construction of quantum codes. Six new families of asymmetric quantum codes derived from constacyclic codes are constructed in this paper. Moreover, the constructed asymmetric quantum codes are optimal and different from the codes available in the literature.

scholarly journals Quantum Codes Obtained through Constacyclic Codes over Z3 + νZ3 + ωZ3 + νωZ3

2021 ◽  
Vol 14 (3) ◽  
pp. 1082-1097
Author(s):  
Jagbir Singh ◽  
Prateek Mor ◽  
Shikha . ◽  
Meena .
Keyword(s):  
Structural Properties ◽  
Gray Map ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Arbitrary Length ◽  
Negacyclic Codes ◽  
Characteristic 3

This paper is concerned with, structural properties and construction of quantum codes over Z3 by using constacyclic codes over the finite commutative non-chain ring R = Z3 + νZ3 + ωZ3 + νωZ3 where ν2 = 1, ω2 = ω, νω = νω, and Z3 is field having 3 elements with characteristic 3. A Gray map is defined between R and Z43. The parameters of quantum codes over Z3 are obtained by decomposing constacyclic codes into cyclic and negacyclic codes over Z3. As an application, some examples of quantum codes of arbitrary length, are also obtained.

Constacyclic codes over the ring Fq[u,v,w]/〈u2 − 1, v2 − 1,w3 − w, uv − vu,vw − wv,wu − uw〉

2021 ◽  
Author(s):  
Shunhua Zhang
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Gray Map ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Direct Sum ◽  
Necessary And Sufficient

Let [Formula: see text] be the ring [Formula: see text], where [Formula: see text] for any odd prime [Formula: see text] and positive integer [Formula: see text]. In this paper, we study constacyclic codes over the ring [Formula: see text]. We define a Gray map by a matrix and decompose a constacyclic code over the ring [Formula: see text] as the direct sum of constacyclic codes over [Formula: see text], we also characterize self-dual constacyclic codes over the ring [Formula: see text] and give necessary and sufficient conditions for constacyclic codes to be dual-containing. As an application, we give a method to construct quantum codes from dual-containing constacyclic codes over the ring [Formula: see text].

Constacyclic codes over mixed alphabets and their applications in constructing new quantum codes

2021 ◽  
Vol 20 (4) ◽  
Author(s):  
Hai Q. Dinh ◽  
Sachin Pathak ◽  
Tushar Bag ◽  
Ashish Kumar Upadhyay ◽  
Woraphon Yamaka
Keyword(s):  
Quantum Codes ◽  
Constacyclic Codes

Quantum codes from nearly self-orthogonal quaternary linear codes

2014 ◽  
Vol 73 (2) ◽  
pp. 417-424 ◽  
Author(s):  
Petr Lisoněk ◽  
Vijaykumar Singh
Keyword(s):  
Linear Codes ◽  
Quantum Codes ◽  
Quaternary Linear Codes

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>

Sign in / Sign up

Export Citation Format

Share Document