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.

We introduce quasi-cyclic codes of index 1(1/3), and construct a class of such codes generated by pairs of polynomials. By investigating the pair of circulant matrices associated with the generator pair of polynomials, we obtain the generator matrix of any code of the class. Using a probabilistic method, we prove that, for any positive real number δ such that the asymptotic GV-bound at 2δ is greater than 1/2, the probability that the relative minimal distance of the code in the class is greater than δ is almost 1; and the probability that the rate of the code equals to 1/4 is also almost 1. An obvious consequence is that the quasicyclic codes of index 1(1/3) are asymptotically good.