Annihilator and twisted Euclidean duality for quasi-polycyclic codes

Let $f\in\mathbb F_q[x]$ be a monic polynomial of degree $m$ with $f(0)\ne 0$, and let $\mathcal R=\mathbb F_q[x]/\langle f\rangle$. Under coefficient expansion, a quasi-polycyclic (QP) code of index $n$ corresponds to an $\mathcal R$-submodule of $\mathcal R^n$. In this paper, we study QP codes wit