Strong Measure Zero Sets on $2^{\kappa }$ for $\kappa$ Inaccessible

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 $$\mathrm{\mid }{2}^{\kappa }\mathrm{\mid }={\kappa }^{++}+\mathrm{\forall }X\subseteq 2^{\kappa }:X\text{ is strong measure zero if and only if }\mathrm{\mid }X\mathrm{\mid }\le {\kappa }^{+}\mathrm{.}$$ 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.