  We study the problem of existence of (nontrivial) perfect codes in the
discrete $ n $-simplex $ \Delta_{\ell}^n := \left\{ \begin{pmatrix} x_0,
\ldots, x_n \end{pmatrix} : x_i \in \mathbb{Z}_{+}, \sum_i x_i = \ell \right\}
$ under $ \ell_1 $ metric. The problem is motivated by the so-called multiset
codes, which have recently been introduced by the authors as appropriate
constructs for error correction in the permutation channels. It is shown that $
e $-perfect codes in the $ 1 $-simplex $ \Delta_{\ell}^1 $ exist for any $ \ell
\geq 2e + 1 $, the $ 2 $-simplex $ \Delta_{\ell}^2 $ admits an $ e $-perfect
code if and only if $ \ell = 3e + 1 $, while there are no perfect codes in
higher-dimensional simplices. In other words, perfect multiset codes exist only
over binary and ternary alphabets.
