  We show that given $n$ and $k$, for $q$ sufficiently large, there always
exists an $[n, k]_q$ MDS code that has a generator matrix $G$ satisfying the
following two conditions: (C1) Sparsest: each row of $G$ has Hamming weight $n
- k + 1$; (C2) Balanced: Hamming weights of the columns of $G$ differ from each
other by at most one.
