Repeated-root constacyclic codes of length 3p ^s over the finite non-chain ring (Formula presented) and their duals

This study aims to determine the algebraic structures of α-constacyclic codes of length 3p ^s over the finite commutative non-chain ring (Formula presented), for a prime p≠3. For the unit α, we consider two different instances: when α is a cube in ℛ and when it is not. Analyzing the first scenario is relatively easy. When α is not a unit in ℛ, we consider several subcases and determine the algebraic structures of constacyclic codes in those cases. Further, we also provide the number of codewords and the duals of α-constacyclic codes.