New nonexistence results on perfect permutation codes under the hamming metric

Permutation codes under the Hamming metric are interesting topics due to their applications in power line communications and block ciphers. In this paper, we study perfect permutation codes in \begin{document}$ S_n $\end{document}, the set of all permutations on \begin{document}$ n $\end{document} elements, under the Hamming metric. We prove the nonexistence of perfect \begin{document}$ t $\end{document}-error-correcting codes in \begin{document}$ S_n $\end{document} under the Hamming metric, for more values of \begin{document}$ n $\end{document} and \begin{document}$ t $\end{document}. Specifically, we propose some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect \begin{document}$ t $\end{document}-error-correcting code in \begin{document}$ S_n $\end{document} under the Hamming metric for some \begin{document}$ n $\end{document} and \begin{document}$ t = 1,2,3,4 $\end{document}, or \begin{document}$ 2t+1\leq n\leq \max\{4t^2e^{-2+1/t}-2,2t+1\} $\end{document} for \begin{document}$ t\geq 2 $\end{document}, or \begin{document}$ \min\{\frac{e}{2}\sqrt{n+2},\lfloor\frac{n-1}{2}\rfloor\}\leq t\leq \lfloor\frac{n-1}{2}\rfloor $\end{document} for \begin{document}$ n\geq 7 $\end{document}, where \begin{document}$ e $\end{document} is the Napier's constant.