New Quantum Codes and (θ, δ, β)-Cyclic Codes

For a prime p and a positive integer e, let
$\mathbb {F}_{p^{e}}$
be the finite field of order
$p^{e}$
and
$\mathfrak {R}_{\ell } $
be a non-chain ring given by
$\mathfrak {R}_{\ell } :=\mathbb {F}_{p^{e}}[v]/\langle v^{\ell } -1\rangle,~\ell \gt 1$
. This study presents the
$(\theta _{1},\delta _{1},\beta _{1})$
-cyclic codes over
$\mathfrak {R}_{\ell } $
and
$(\theta _{i},\delta _{i},\beta _{i})$
-cyclic codes over
$\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell } $
for
$i=0,1$
. Further, we study the application of these codes in the construction of quantum codes. Towards this, we define a Gray map to find images of the elements of
$\mathfrak {R}_{\ell } $
and
$\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell } $
over the copies of
$\mathbb {F}_{p^{e}}$
, 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.