The GM-MDS conjecture for constrained generator matrices

Let $M$ be an $m\times n$ binary support matrix. For a nonempty set $I\subseteq\{1,\dots,m\}$, write $M_i$ for the $i$th row and $\operatorname{supp}(M_i)$ for its support. The matrix $M$ satisfies the MDS condition when

$$\left|\bigcup_{i\in I}\operatorname{supp}(M_i)\right|\geq n-m+|I|.$$

An $[n,m]_q$ code is MDS when every $m\times m$ submatrix of a generator matrix is full-rank, and a generator matrix $G$ fits $M$ when $M_{i,j}=0$ implies $G_{i,j}=0$. GM-MDS conjecture. If $M$ satisfies the MDS condition, then for any field $\mathbb{F}$ of size $q\geq n+m-1$, there exists an $[n,m]_q$ MDS code whose generator matrix $G$ with entries in $\mathbb{F}$ fits the matrix $M$. This conjecture seeks a small-field completion theorem for constrained MDS generator matrices; it was still open in general in the source, although several special cases were known.