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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 non-zero.

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Do you mean "zero" at the end? –  Brendan McKay Dec 5 '12 at 8:31
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Do you really want your collection of vectors `$\{x_1,\ldots, x_n\}$' to be unordered? –  Rudi Pendavingh Dec 5 '12 at 9:29
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Siegel's lemma will give you an estimate. –  Felipe Voloch Dec 5 '12 at 16:16
    
Yes, I did mean "zero" at the end. –  TOM Feb 17 '13 at 12:43
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