Weight distributions of all irreducible $\mu$-constacyclic codes of length ${\mathrm{\ell }}^n$

Let ${\mathbb{F}}_q$ be a finite field of order $q$ and integer $n\ge 1$. Let $\mathrm{\ell }$ be a prime such that ${\mathrm{\ell }}^k\mathrm{\mid }(q-1)$ for some integer $k\ge 1$ and $\mu$ be an element of order ${\mathrm{\ell }}^k$ in ${\mathbb{F}}_q$. In this paper, we determine the weight distributions of all irreducible $\mu$-constacyclic codes of length ${\mathrm{\ell }}^n$ over ${\mathbb{F}}_q$. Explicit expressions for the generator polynomials and codewords of these codes are also obtained.