Hi,

I want to whether there is a vector generating function (/matrix) such that it can generate a m-dimensional vector which will always be linearly independent of the set of m-dimensional vectors the function has already generated.

My problem can be written in pseudocode format as follow. I therefore expect that any m randomly picked vectors from the pool of the N vectors will generate a full-rank matrix.

```
For (n=1; n<N; n++) { //N>m
S = Span (v1, v2, ..., vn-1)
//i.e. vn is linearly independent of the set of vectors already generated.
Generate vector vn, such that vn is not an element of S;
S = Span (v1, v2, ..., vn)
}
```

Vandermonde matrix is one possible option, but it requires the use of exponentially large field size. So I am looking for vectors generated over smaller field size. Any help in this direction will be greatly appreciated.

If this is an open research problem, then please do advice me.

Thanks in advance.