Perfect codes in the discrete simplex

Abstract

We study the problem of existence of (nontrivial) perfect codes in the discrete $n$-simplex $\Delta _{\ell }^n := \left\{ \left( \begin{array}{l} x_0, \ldots , x_n \end{array}\right) : x_i \in {\mathbb {Z}}_{+}, \sum _i x_i = \ell \right\}$ 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 $\ell \ge 2e + 1$, 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.