I have an $m \times n$ matrix $M$ where $m < n$ over finite field of size $2^w$. This matrix has the property that every $m \times m$ matrix formed by a $m$-subset of its columns is invertible. Is there a way to extend $M$ into a $m \times (n + k)$ matrix while still preserving this property?
The motivation behind this problem is for erasure coding in coding theory, where $M$ is the generator matrix. I want to extend the code to add more parity bits without needing to re-encode the existing parity bits.