Let $\mathbb{F}_q$ be a finite field of $q=p^m$ elements where $p$ is a prime and $m$ is a positive integer. This paper considers $(γ,Δ)$-cyclic codes over a class of finite non-chain commutative rings $\mathscr{R}_{q,s}=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle v_i-v_i^2,v_iv_j=v_jv_i=0\rangle$ where $γ$ is an automorphism of $\mathscr{R}_{q,s}$, $Δ$ is a $γ$-derivation of $\mathscr{R}_{q,s}$ and $1\leq i\neq j\leq s$ for a positive integer $s$. Here, we show that a $(γ,Δ)$-cyclic code of length $n$ over $\mathscr{R}_{q,s}$ is the direct sum of $(θ,\Im)$-cyclic codes of length $n$ over $\mathbb{F}_q$, where $θ$ is an automorphism of $\mathbb{F}_q$ and $\Im$ is a $θ$-derivation of $\mathbb{F}_q$. Further, necessary and sufficient conditions for both $(γ,Δ)$-cyclic and $(θ,\Im)$-cyclic codes to contain their Euclidean duals are established. Then, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. These codes have better parameters than those available in the literature. Finally, the encoding and error-correction procedures for our proposed quantum codes are discussed.
Encoding and Construction of Quantum Codes from (γ,Δ)-cyclic Codes over a Class of Non-chain Rings
Let Fq be a finite field of q=pm elements where p is a prime and m is a positive integer. This paper considers (γ,Δ)-cyclic codes over a class of finite non-chain commutative rings Rq,s=Fq[v1,v2,…,vs]/⟨vi−vi2,vivj=vjvi=0⟩ where γ is an automorphism of Rq,s, Δ is a γ-derivation of Rq,s and 1≤i=j≤s for a positive integer s. Here, we show that a (γ,Δ)-cyclic code of length n over Rq,s is the direct sum of (θ,ℑ)-cyclic codes of length n over Fq, where θ is an automorphism of Fq and ℑ is a θ-derivation of Fq. Further, necessary and sufficient conditions for both (γ,Δ)-cyclic and (θ,ℑ)-cyclic codes to contain their Euclidean duals are established. Then, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. These codes have better parameters than those available in the literature. Finally, the encoding and error-correction procedures for our proposed quantum codes are discussed.
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.
(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.
In this paper, we extend the results given in [3] to a nonchain ring Rp=Fp+vFp+⋯+vp−1Fp , where vp=v and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring Rp and study the structure of their duals. We classify cyclic codes containing their duals over Rp by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map π defined in [4], we give the parameters of the quantum codes of length pn over Fp which are obtained from cyclic codes over Rp . Finally, we illustrate the results by giving some examples.
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.
Quantum codes constructed from cyclic codes over the ring Fq+vFq+v2Fq+v3Fq+v4Fq
In this article, we investigate properties of cyclic codes over a finite non-chain ring Fq+vFq+v2Fq+v3Fq+v4Fq, where q=pr,r is a positive integer, p is an odd prime, 4∣(p−1), and v5=v. As an application, we construct several quantum error correcting codes over the finite field Fq.