Let $p$ be a prime and $\mathbb{F}_q$ be the finite field of order $q=p^m$. In this paper, we study $\mathbb{F}_q\mathcal{R}$-skew cyclic codes where $\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_q$ with $u^2=u$. To characterize $\mathbb{F}_q\mathcal{R}$-skew cyclic codes, we first establish their algebraic structure and then discuss the dual-containing properties by considering a non-degenerate inner product. Further, we define a Gray map over $\mathbb{F}_q\mathcal{R}$ and obtain their $\mathbb{F}_q$-Gray images. As an application, we apply the CSS (Calderbank-Shor-Steane) construction on Gray images of dual containing $\mathbb{F}_q\mathcal{R}$-skew cyclic codes and obtain many quantum codes with better parameters than the best-known codes available in the literature.
FqR-skew cyclic codes and their application to quantum codes
Let p be a prime and Fq be the finite field of order q=pm. In this paper, we study FqR-skew cyclic codes where R=Fq+uFq with u2=u. To characterize FqR-skew cyclic codes, we first establish their algebraic structure and then discuss the dual-containing properties by considering a non-degenerate inner product. Further, we define a Gray map over FqR and obtain their Fq-Gray images. As an application, we apply the CSS (Calderbank-Shor-Steane) construction on Gray images of dual containing FqR-skew cyclic codes and obtain many quantum codes with better parameters than the best-known codes available in the literature.
For an odd prime $ p $ and positive integers $ m $ and $ \ell $, let $ \mathbb{F}_{p^m} $ be the finite field with $ p^{m} $ elements and $ 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} $. Thus $ R_{\ell,m} $ is a finite commutative non-chain ring of order $ p^{2^{\ell} m} $ with characteristic $ p $. In this paper, we aim to construct quantum codes from skew constacyclic codes over $ R_{\ell,m} $. 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.
(Θ,ΔΘ,a)-cyclic codes over Fql and their applications in the construction of quantum codes
In this article, for a finite field Fq and a natural number l, let R denote the product ring Fql. Firstly, for an automorphism Θ of R, a Θ-derivation ΔΘ of R and for a unit a in R, we study (Θ,ΔΘ,a)-cyclic codes over R. In this direction, we give an algebraic characterization of a (Θ,ΔΘ,a)-cyclic code over R, determine its generator polynomial, and find its decomposition over Fq. Secondly, we give a necessary and sufficient condition for a (Θ,0,a)-cyclic code to be Euclidean dual-containing code over R. Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over Fq as Gray images of (Θ,ΔΘ,a)-cyclic codes over R. Moreover, we determine orthogonality preserving Gray maps and construct Euclidean dual-containing codes with good parameters. Lastly, as an application, we construct MDS and almost MDS quantum codes by employing the Euclidean dual-containing and annihilator dual-containing CSS constructions.
In this article, for a finite field $ \mathbb{F}_q $ and a natural number $ l, $ let $ \mathcal{R} $ denote the product ring $ \mathbb{F}_q^l. $ Firstly, for an automorphism $ \Theta $ of $ \mathcal{R}, $ a $ \Theta $-derivation $ \Delta_\Theta $ of $ \mathcal{R} $ and for a unit $ \mathit{\boldsymbol{a}} $ in $ \mathcal{R}, $ we study $ (\Theta, \Delta_\Theta, \mathit{\boldsymbol{a}}) $-cyclic codes over $ \mathcal{R}. $ In this direction, we give an algebraic characterization of a $ (\Theta, \Delta_\Theta, \mathit{\boldsymbol{a}}) $-cyclic code over $ \mathcal{R} $, determine its generator polynomial, and find its decomposition over $ \mathbb{F}_q. $ Secondly, we give a necessary and sufficient condition for a $ (\Theta, 0, \mathit{\boldsymbol{a}}) $-cyclic code to be Euclidean dual-containing code over $ \mathcal{R}. $ Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over $ \mathbb{F}_q $ as Gray images of $ (\Theta, \Delta_\Theta, \mathit{\boldsymbol{a}}) $-cyclic codes over $ \mathcal{R}. $ Moreover, we determine orthogonality preserving Gray maps and construct Euclidean dual-containing codes with good parameters. Lastly, as an application, we construct MDS and almost MDS quantum codes by employing the Euclidean dual-containing and annihilator dual-containing CSS constructions.
New non-binary quantum codes from skew constacyclic codes over the ring Fpm+vFpm+v2Fpm
In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring R=Fpm[v]/⟨v3=v⟩ where p is an odd prime and m≥1. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring R. Then a necessary and sufficient condition for skew constacyclic codes over R to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over Fpm.
In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring R=Fpm[v]/⟨v3=v⟩ where p is an odd prime and m≥1. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring R. Then a necessary and sufficient condition for skew constacyclic codes over R to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over Fpm.
For a prime p and a positive integer e, letFpebe the finite field of orderpeand$\mathfrak {R}_{\ell } beanon−chainringgivenby\mathfrak {R}_{\ell } :=\mathbb {F}_{p^{e}}[v]/\langle v^{\ell } -1\rangle,~\ell \gt 1.Thisstudypresentsthe(\theta _{1},\delta _{1},\beta _{1})−cycliccodesover\mathfrak {R}_{\ell } and(\theta _{i},\delta _{i},\beta _{i})−cycliccodesover\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell } fori=0,1.Further,westudytheapplicationofthesecodesintheconstructionofquantumcodes.Towardsthis,wedefineaGraymaptofindimagesoftheelementsof\mathfrak {R}_{\ell } and\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell } overthecopiesof\mathbb {F}_{p^{e}}$, and then establish the dual-containing conditions for these codes. Finally, using the CSS construction and a propagation rule, we find many non-binary quantum and classical codes with better parameters.