On extended perfect codes

We consider extended $1$-perfect codes in Hamming graphs $H(n,q)$. Such nontrivial codes are known only when $n=2^k$, $k\ge 1$, $q=2$, or $n=q+2$, $q=2^m$, $m\ge 1$. Recently, Bespalov proved nonexistence of extended $1$-perfect codes for $q=3$, $4$, $n>q+2$. In this work, we characterize all positive integers $n$, $r$ and prime $p$, for which there exist such a code in $H(n,p^r)$. We also consider $2$-perfect codes in Hamming $H(n,q)$ and Johnson graphs $J(n,w)$ and find new necessary conditions on there existence.