Minimal subshifts of prescribed mean dimension over general alphabets

Let $G$ be a countable infinite amenable group, $K$ a finite-dimensional compact metrizable space, and $(K^G,\sigma )$ the full $G$-shift on $K^G$. For any $r\in [0,\mathrm{m}\mathrm{d}\mathrm{i}\mathrm{m}(K^G,\sigma ))$, we construct a minimal subshift $(X,\sigma )$ of $(K^G,\sigma )$ with mdim$(X,\sigma )=r$. Furthermore, we construct a subshift of $([0,1{]}^G,\sigma )$ such that its mean dimension is $1$, and that the set of all attainable values of the mean dimension of its minimal subsystems is exactly the interval $[0,1)$.