Recently, many good quantum codes over various finite fields $F_q$ have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from cyclic codes or their generalizations over $F_q$. Unless explicit benefits are demonstrated for the indirect approach, we believe that direct and more elementary methods should be preferred.
Recently, many good quantum codes over various finite fields Fq have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from cyclic codes or their generalizations over Fq. Unless explicit benefits are demonstrated for the indirect approach, we believe that direct and more elementary methods should be preferred.
Building large-scale quantum computers is a very active area of research. A critical part of this effort is the ability to control quantum errors. Recently, many good quantum error correcting codes (QECC) over various finite fields Fq have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from classical codes, including cyclic, constacyclic, quasicyclic (QC), quasi-twisted (QT), and polycyclic codes using the CSS construction, Hermitian construction, and Lisonek-Singh methods. We believe direct and more elementary methods should be preferred.
The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of FqPQ, where P=Fq[v]⟨v3−α22v⟩ and Q=Fq[u,v]⟨u2−α12,v3−α22v⟩ are nonchain finite rings and αi is in Fq/{0} for i∈{1,2}, where q=pm with m≥1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes.
(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.
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.
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.