  Let $0<\ell\in\mathbb{Z}$. The notion of an efficient dominating set or
perfect code $S$ of a graph $G$ is generalized to that of an efficient
dominating$\,^\ell$-set or perfect$^\ell$code, of the graph $G$, meaning that
each vertex $v$ of $V(G)\setminus S$ has exactly $\ell$ neighbors in $S$,
instead of just one neighbor. Such generalization is applied to star $j$-set
transposition graphs based on permutations of multisets with each symbol
repeated $j$ times, ($j\in\{\ell,\ell-1\}$). In such vertex-transitive graphs
this approach produces total colorings, efficient dominating sets, also called
perfect codes, etc.
