Intersections of $q$-ary perfect codes

The intersections of $q$-ary perfect codes are under study. We prove that there exist two $q$-ary perfect codes $C_1$ and $C_2$ of length $N=qn+1$ such that $|C_1\cap C_2|=k\cdot|P_i|/p$ for each $k\in\{0,\dots,p\cdot K-2,p\cdot K\}$, where $q=p^r$, $p$ is prime, $r\ge1$, $n=\dfrac{q^{m-1}-1}{q-1}$, $m\ge2$, $|P_i|=p^{nr(q-2)+n}$ and $K=p^{n(2r-1)-r(m-1)}$. We show also that there exist two $q$-ary perfect codes of length $N$ which are intersected by $p^{nr(q-3)+n}$ codewords.