  A multifold $1$-perfect code ($1$-perfect code for list decoding) in any
graph is a set $C$ of vertices such that every vertex of the graph is at
distance not more than $1$ from exactly $\mu$ elements of $C$. In $q$-ary
Hamming graphs, where $q$ is a prime power, we characterize all parameters of
multifold $1$-perfect codes and all parameters of additive multifold
$1$-perfect codes. In particular, we show that additive multifold $1$-perfect
codes are related to special multiset generalizations of spreads, multispreads,
and that multispreads of parameters corresponding to multifold $1$-perfect
codes always exist.
  Keywords: perfect codes, multifold packing, multiple covering, list-decoding
codes, additive codes, spreads, multispreads, completely regular codes,
intriguing sets.
