Let $\mathbb{F}_{2^\sigma}$ be a finite field of $\sigma$-bit elements, and use $\mathbb{F}_{2^\sigma}^{\ell}$ to denote an $\ell$-dimensional vector space over $\mathbb{F}_{2^\sigma}$. Let $V$ be a set of vectors in $\mathbb{F}_{2^\sigma}^\ell$ such that any $\ell$ vectors in $V$ are linearly independent. The question are:
- Is it possible that $|V|>\ell$?
- How to find such $V$ of maximum size?
$\big($The questions are not very hard to answer for vectors over real numbers $\mathbb{R}$. In that case, "yes" to the first question, and $V=\{e_1,\dots,e_{\ell}\}\cup\{[1, x,\dots,x^{2^\sigma-1}]| x\in\mathbb{F}_{2^\sigma}\}$. So $|V|=2^\sigma+(\ell-1)$.$\big)$
