Double Constacyclic Codes Over Two Finite Commutative Chain Rings

Many kinds of codes which possess two cycle structures over two special finite commutative chain rings, such as
${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$
-additive cyclic codes and quasi-cyclic codes of fractional index etc., were proved asymptotically good. In this paper we extend the study in two directions: we consider any two finite commutative chain rings with a surjective homomorphism from one to the other, and consider double constacyclic structures. We construct an extensive kind of double constacyclic codes over two finite commutative chain rings. And, developing a probabilistic method suitable for quasi-cyclic codes over fields, we prove that the double constacyclic codes over two finite commutative chain rings are asymptotically good.