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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:

  1. Is it possible that $|V|>\ell$?
  2. 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 $|V|=2^\sigma+(\ell-1)$.$\big)$

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  • $\begingroup$ I don't understand your example of a set $V$. Doesn't your notation imply that $2^\sigma$ equals $l$ in this case? Also why do you write 'In that case' if the example is clearly not over the reals? $\endgroup$
    – Vincent
    Commented Sep 29, 2015 at 22:26
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    $\begingroup$ You are looking for the main conjecture for MDS codes. It is possible that $|V| > \ell$. $\endgroup$
    – Felipe Voloch
    Commented Sep 30, 2015 at 1:24

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