For notation, I will use $p$ instead of $p^n$, and just say $p$ is a prime power. I believe $|K|=k$ where $k$ is the smallest integer such that
For , I find . For , . I haven't checked for larger yet.
Let $V$ be an $m$-dimensional vector space over a finite field of order $p^n$ for some prime $p$ and some positive integer $n$. Let $K\subset V$ be a collection of vectors such that every subset $K'\subset K$ with $|K'|=m$ is a basis for $V$. What is the maximum size of $|K|$?
For notation, I will use $p$ instead of $p^n$, and just say $p$ is a prime power. I believe $|K|=k$ where $k$ is the smallest integer such that
For , I find . For , . I haven't checked for larger yet.
I'm wrong. For , .
I know why I was wrong. Suppose and we want to add one additional vector. Let's figure out the set of forbidden vectors. The zero vector is forbidden. There are nonzero multiples for each of the vectors. There are linear combinations of each of the pairs of vectors. So, the number of forbidden vectors is
So, we want the smallest positive integer such that
For , that means , so means , or .
For , that means
Simplifying, we have
So,
Is that right? Or is there a possibility for overlap? Are linear combinations of 2 vectors unique among other sets of 2 vectors where among the four vectors, any three are linearly independent?