Permutation codes are nonempty subsets of the symmetric group $S_n$ of all permutations over $\{1, 2,\dots , n\}$. These codes have been studied under various metrics, as they have nontrivial applications in flash memories. In this paper, the authors focus on permutation codes under the Kendall $\tau$-metric, which measures the minimum number of adjacent transpositions required to obtain the permutation. The main result of the paper is the proof that perfect codes under this metric do not exist for some choices of the parameters, which extends some classical results for codes over other metrics.