On extended perfect codes
<|reference_start|>On extended perfect codes: We consider extended $1$-perfect codes in Hamming graphs $H(n,q)$. Such nontrivial codes are known only when $n=2^k$, $k\geq 1$, $q=2$, or $n=q+2$, $q=2^m$, $m\geq 1$. 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 $H(n,p^r)$.<|reference_end|>
@article{vorob'ev2024on,
    title={On extended perfect codes},
    author={Konstantin Vorob'ev},
    journal={arXiv preprint arXiv:2403.10992},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.10992},
    primaryClass={math.CO cs.IT math.IT}
}