Open AccessNew nonexistence results on perfect permutation codes under the hamming metric
Author(s) -
Xiang Wang,
Wenjuan Yin
Publication year - 2023
Publication title -
advances in mathematics of communications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.601
H-Index - 26
eISSN - 1930-5346
pISSN - 1930-5338
DOI - 10.3934/amc.2021058
Subject(s) - mathematics , permutation (music) , combinatorics , hamming bound , hamming distance , hamming code , metric (unit) , discrete mathematics , block code , algorithm , physics , decoding methods , acoustics , operations management , economics
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.