Repeated-root constacyclic codes of length $p_1{p}_2^t{p}^s$ and their dual codes

Let ${\mathbb{F}}_q$ be the finite field with $q=p^k$ elements, and $p_1,p_2$ be two distinct prime numbers different from $p$. In this paper, we first calculate all the $q$-cyclotomic cosets modulo $p_1{p}_2^t$ as a preparation for the following parts. Then we give the explicit generator polynomials of all the constacyclic codes of length $p_1{p}_2^t{p}^s$ over ${\mathbb{F}}_q$ and their dual codes. In the rest of this paper, we determine all self-dual cyclic codes of length $p_1{p}_2^t{p}^s$ and their enumeration. This answers a question recently asked by B. Chen, H.Q.Dinh and Liu. In the last section, we calculate the case of length $5\ell p^s$ as an example.