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Denote by $\\omega_1$ the $p^{th}$ primitive unity root, $\\omega_1:=e^{\\frac{2\\pi i}{p}}$. Define $\\omega_i:=\\omega_1^i$ for $0\\leq i\\leq p-1$ and $B:=\\{1,\\omega_1,...,\\omega_{p-1}\\}^n$. Denote by $K(n,p)$ the minimum $k$ for which there exist vectors $v_1,...,v_k\\in B$ such that for any vector $w\\in B$, there is an $i$, $1\\leq i\\leq k$, such that $v_i\\cdot w=0$, where $v\\cdot w$ is the usual scalar product of $v$ and $w$. Gr\\\"obner basis methods and linear algebra proof gives the lower bound $K(n,p)\\geq n(p-1)$. 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