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, let
Fpebe the finite field of order
peand$\mathfrak {R}_{\ell }
beanon−chainringgivenby\mathfrak {R}_{\ell } :=\mathbb {F}_{p^{e}}[v]/\langle v^{\ell } -1\rangle,~\ell \gt 1
.Thisstudypresentsthe(\theta _{1},\delta _{1},\beta _{1})
−cycliccodesover\mathfrak {R}_{\ell }
and(\theta _{i},\delta _{i},\beta _{i})
−cycliccodesover\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell }
fori=0,1
.Further,westudytheapplicationofthesecodesintheconstructionofquantumcodes.Towardsthis,wedefineaGraymaptofindimagesoftheelementsof\mathfrak {R}_{\ell }
and\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell }
overthecopiesof\mathbb {F}_{p^{e}}$, 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.