What is the minimal $m$ such that there exists a set $A = \{a_1,...a_n\}$ of vectors : $a_i \in \{0,1\}^m$ ($n$ is given) such that every subset of vectors of size $k$ is independent, but only with scalars $\alpha_i = \{-1,0,1\}$? What if we also require that $||a_i||_0 = s$? Is there a way to find such a set?
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$\begingroup$ are you working over $\mathbb{R}$? $\endgroup$– kodluCommented Mar 27, 2020 at 8:59
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$\begingroup$ Edited, basically an answer over $\mathbb{R}$ would be ok, but I am interested in the case where all the coefficients of the linear combination can only be $-1,1,0$ $\endgroup$– SomeoneHAHACommented Mar 27, 2020 at 10:32
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$\begingroup$ Take $a_i=e_i$, where $e_i$ is the standard vector basis. $\endgroup$– ShahroozCommented Mar 27, 2020 at 11:37
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