Permutation codes are studied due to their numerous applications in various applications, such as power line communications, block ciphers, and coding for storage. In this paper, we study perfect permutation codes in \(S_n\), the set of all permutations on n elements, under the \(\ell _{\infty }\)-metric. We present some nonexistence results on perfect t-error-correcting permutation codes in \(S_n\) under the \(\ell _{\infty }\)-metric for some t and n. More precisely, we prove that there does not exist a perfect t-error-correcting code in \(S_n\) under the \(\ell _{\infty }\)-metric for t and n satisfying \(1\le t\le 3, 2t+1\le n\) or for t and n satisfying \(R_{2t+1}(n)=0,1,2t\), where \(0\le R_d(n)< d\) is the residue when dividing the positive integer n by the positive integer d.

置换码因其在各种应用中的大量应用而被研究，例如电力线通信、分组密码和存储编码。在本文中，我们在\(\ell _{\infty }\)度量下研究了\(S_n\)中的完美置换码，它是n 个元素上所有排列的集合。对于某些t和n ，我们在\(\ell _{\infty }\)度量下的\ (S_n\)中提出了一些关于完美t纠错置换码的不存在结果。更准确地说，我们证明在\(S_n\)中不存在完美的t纠错码\(\ell _{\infty }\) - t和n满足\(1\le t\le 3, 2t+1\le n\)或t和n满足\(R_{2t+1} (n)=0,1,2t\)，其中\(0\le R_d(n)< d\)是正整数n除以正整数d时的余数。