  We investigate the notion of strong measure zero sets in the context of the
higher Cantor space $2^\kappa$ for $\kappa$ at least inaccessible. Using an
iteration of perfect tree forcings, we give two proofs of the relative
consistency of \[ |2^\kappa| = \kappa^{++} + \forall X \subseteq 2^\kappa:\ X
\text{ is strong measure zero if and only if } |X| \leq \kappa^+. \]
Furthermore, we also investigate the stronger notion of stationary strong
measure zero and show that the equivalence of the two notions is undecidable in
ZFC.
