Perfect codes in the discrete simplex

We study the problem of existence of (nontrivial) perfect codes in the discrete n-simplex ∆` := {( x0, . . . , xn ) : xi ∈ Z+, ∑ i xi = ` } under `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 ∆` exist for any ` ≥ 2e + 1, the 2-simplex ∆ 2 ` admits an e-perfect code if and only if ` = 3e + 1, while there are no perfect codes in higherdimensional simplices. In other words, perfect multiset codes exist only over binary and ternary alphabets.