The weight distributions of constacyclic codes

Let \begin{document}$\Bbb F_q$\end{document} be a finite field with \begin{document}$q$\end{document} elements. Suppose that \begin{document}$a, λ∈ \Bbb F_q^*$\end{document} , \begin{document}$a^n=λ$\end{document} with \begin{document}$n|(q-1)$\end{document} . In this paper, we determine the weight distribution of a class of \begin{document}$λ$\end{document} -constacyclic codes of length \begin{document}$nm$\end{document} with the parity check polynomial \begin{document}$h(x)=(x^m-aξ^{st})(x^m-aξ^{s(t+1)})...(x^m-aξ^{s(t+r-1)})$\end{document} and \begin{document}$n>(r-1)m$\end{document} , where \begin{document}$s,t, r$\end{document} are positive integers and \begin{document}$ξ∈ \Bbb F_q$\end{document} is a primitive n-th root of unity. Moreover, we give the weight distributions of \begin{document}$λ$\end{document} -constacyclic codes of length \begin{document}$nm$\end{document} explicitly in several cases: (1) \begin{document}$r=1$\end{document} , \begin{document}$n>1$\end{document} ; (2) \begin{document}$r=2$\end{document} , \begin{document}$m=2$\end{document} and \begin{document}$n>2$\end{document} ; (3) \begin{document}$r=2$\end{document} , \begin{document}$m=3$\end{document} and \begin{document}$n>3$\end{document} ; (4) \begin{document}$r=3$\end{document} , \begin{document}$m=2$\end{document} and \begin{document}$n>4$\end{document} .