Perfect Codes in the Discrete Simplex

Abstract: 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.