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More formally, if $\\vec{u}_1,\\ldots,\\vec{u}_t \\in \\Z_m^n$ and $\\vec{v}_1,\\ldots,\\vec{v}_t \\in \\Z_m^n$ satisfy $\\langle\\vec{u}_i,\\vec{v}_i\\rangle\\equiv0\\pmod m$ and $\\langle\\vec{u}_i,\\vec{v}_j\\rangle\\not\\equiv0\\pmod m$ for all $i\\neq j\\in[t]$, we prove that $t \\leq O(m^{n/2+8.47})$. 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