Spaces for Which the Generalized Cantor Space $2^J$ is a Remainder

Let $X$ be a locally compact noncompact space, $m$ be an infinite cardinal and $|J| = m$. Let $F(X)$ be the algebra of continuous functions from $X$ into $\mathbf{R}$ which have finite range outside of an open set with compact closure and let $I(X) = \{g \in F(X): g\quad\text{vanishes outside of an open set with compact closure}\}$. Conditions on $R(X) = F(X)/I(X)$ and internal conditions are obtained which characterize when $X$ has $2^J$ as a remainder.