Suppose we have $n$ strings (or vectors) $a_1, a_2, \dots, a_n \in A^m$, where $A$ is an arbitrary set satisfies $|A| \geq n$. And we limit their pairwise hamming distance by $$ d(a_i, a_j) \geq d_{ij} \mathrm{\ for\ all\ } i \neq j$$ with given $d_{ij}$. Then we define random variable $X_l$ by that it will equal to $a_{il}$ (the $l$-th character, or component, of $a_i$) for some $i$, where $i$ will be uniformly distributed in $1,2,\dots,n$.

Question: Can we give a lower bound of $$ E(H(X_l)) = \frac{1}{m}\sum_{l=1}^m H(X_m),$$ where $H(\cdot)$ is entropy function and $l$ is uniform distributed in $1,2,\dots,m$?