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Let $V_n=\{-1,1\}^n$ be the hypercube and let $C_n$ be a collection $\{x_1,...,x_n\}$ of $n$ distinct elements of $V_n.$

Question: what is the smallest number $N(n)$ of non-zero vectors with integer coefficients are needed to check that $C_n$ is linearly independent over the integers? That is, what is the smallest set of such vectors $\{v_1,...v_{N(n)}\}$ that if $C_n$ is linearly dependent over the integers then there exists some element $v_{i}$ with coordinates $(k_1,...,k_n)$ such that $k_1x_1+...+k_nx_n$ is zero.

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    $\begingroup$ Do you mean "zero" at the end? $\endgroup$ – Brendan McKay Dec 5 '12 at 8:31
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    $\begingroup$ Do you really want your collection of vectors `$\{x_1,\ldots, x_n\}$' to be unordered? $\endgroup$ – Rudi Pendavingh Dec 5 '12 at 9:29
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    $\begingroup$ Siegel's lemma will give you an estimate. $\endgroup$ – Felipe Voloch Dec 5 '12 at 16:16
  • $\begingroup$ Yes, I did mean "zero" at the end. $\endgroup$ – TOM Feb 17 '13 at 12:43
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A natural idea would be to take all vectors from $[-t,t]^n$ for some $t$. The optimal value of such a $t$ is roughly $n^{n/2}$. This follows from results in this paper: https://link.springer.com/article/10.1007/s10107-011-0474-y.

Can anyone give a better upper bound than $N\le n^{n^2/2}$?

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  • $\begingroup$ I would like to see a set C that is linearly dependent and yet has all 3^n quantities v*C nonzero. Here v ranges over vectors whose coordinates are -1 or 1 or 0. Gerhard "Maybe We Can Do Better" Paseman, 2018.12.05. $\endgroup$ – Gerhard Paseman Dec 5 '18 at 23:23

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