New Non-Binary Quantum Codes from Cyclic Codes Over Product Rings

For any odd prime p, and a divisor ℓ of p, we consider Ie to be the set of all divisors of p - 1, which are less than or equal to ℓ. In this letter, we construct quantum codes from cyclic codes over F
_p
and F
_p
S
_ℓ
, where S
_ℓ
= Π
_i∈Iℓ
R
_i
, for R
_i
= (Fp[u])/(〈u
ⁱ⁺¹
-u〉). For that, first we construct linear codes and a Gray map over R
_ℓ
. Using this construction, we study cyclic codes over R
_ℓ
, and then extend that over F
_p
S
_ℓ
. We also give a Gray map over F
_p
S
_ℓ
. Then, using necessary and sufficient condition of dual containing property for cyclic codes, we construct quantum MDS codes. It is observed that the quantum codes constructed are new in the literature.