On the volume set of point sets in vector spaces over finite fields

Details On the volume set of point sets in vector spaces over finite fields On the volume set of point sets in vector spaces over finite fields

2009-03-13

arxiv.org/abs/0903.2510v1

Mathematics

Combinatorics

Scientific paper

We show that if $\mathcal{E}$ is a subset of the $d$-dimensional vector space over a finite field $\mathbbm{F}_q$ ($d \geq 3$) of cardinality $|\mathcal{E}| \geq (d-1)q^{d - 1}$, then the set of volumes of $d$-dimensional parallelepipeds determined by $\mathcal{E}$ covers $\mathbbm{F}_q$. This bound is sharp up to a factor of $(d-1)$ as taking $\mathcal{E}$ to be a $(d - 1)$-hyperplane through the origin shows.

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