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In this paper, we provide two functions $\\underline{s}(m,n)\\le \\bar{s}(m,n)$ such that the $s^{\\mathrm{th}}$ secant variety of $X_{m,n}$ has the expected dimension if $s \\leq \\underline{s}(m,n)$ or $ \\bar{s}(m,n) \\leq s$. 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