We study the problem of existence of (nontrivial) perfect codes in the discrete n-simplex Δℓn:={(x0,…,xn):xi∈Z+,∑ixi=ℓ}
Nearby in the stack
under
ℓ1
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
Δℓ1
exist for any
ℓ≥2e+1
, the
2
-simplex
Δℓ2
admits an
e
-perfect code if and only if
ℓ=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.