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Given an ordered triple of positive integers $(n,r,b)$, where $1\leqb\leq\binom{n}{r}$, does there exist a matrix of size $r\times n$ with exactly$b$ invertible submatrices of size $r\times r$? Such a matrix is called an$(n,r,b)$-matrix. This question is a stronger version of an open problem inmatroid theory raised by Dominic Welsh. In this paper, we prove that an$(n,r,b)$-matrix exists when the corank satisfies $n-r\leq3$, unless$(n,r,b)=(6,3,11)$. Furthermore, we show that an $(n,r,b)$-matrix exists whenthe rank $r$ is large relative to the corank $n-r$.

Building matrices with prescribed size and number of invertible submatrices