$\newcommand{\F}{{\mathbb F}}$ Let's focus on the case $q=2$. Suppose that $v_1,...,v_m\in\F^n$. Let $V$ be the matrix over $\F_2$ with $v_1,...,v_m$ as its columns. Then $v_1,...,v_m$ are $k$-wise linearly independent if and only if the kernel of $V$ does not contain any vector of weight at most $k$. Therefore, your question is equivalent to, arguably, the central problem of coding theory: what is the largest possible dimension of a linear code which does with the minimal distance at least $k+1$?
Seva
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