On the quantization dimension of maximal linked systems

We prove that for a compact metric space $X$ and for a nonnegative real $b$ not exceeding the lower box dimension of $X$, there exists a maximal linked system in $\lambda X$ with lower quantization dimension $b$ and support $X$. There also exists a maximal linked system in $\lambda X$ with support $X$ whose lower and upper quantization dimensions coincide respectively with the lower and upper box dimensions of $X$.