We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the ‘supertropical semifield’ to the standard tropical case. Using a counting approach, we prove the existence of a $01$-matrix with many ones and without large all-one submatrices, and we put our results together and construct an $ntimes n$ matrix with tropical rank $o(n^{0.5+varepsilon })$ and Kapranov rank $n-o(n)$.
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