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arXiv · 2407.14273

On the cardinality of matrices with prescribed rank and partial trace over a finite field

Kumar Balasubramanian·Krishna Kaipa·Himanshi Khurana
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Abstract

Let $F$ be the finite field of order $q$ and $\M(n,r, F)$ be the set of $n\times n$ matrices of rank $r$ over the field $F$. For $\alpha\in F$ and $A\in \M(n,F)$, let $$Z^{\alpha}_{A,r}=\left\{X\in \M(n,r, F)\mid \tr(AX)=\alpha\right \}.$$ In this article, we solve the problem of determining the cardinality of $Z_{A,r}^{\alpha}$. We also solve the generalization of the problem to rectangular matrices.

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Kumar Balasubramanian, Krishna Kaipa, Himanshi Khurana. 2024-07-19. On the cardinality of matrices with prescribed rank and partial trace over a finite field. https://arxiv.org/abs/2407.14273

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