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I think this is a basic question, but I am not very knowledgeable about the topic.

I am trying to understand this answer to a question of mine.

I know from this answer that the number of hyperplanes in a vector space over a finite field $\mathbb{F}_q^n$ should be $q^{n - k}{n \brack k}_q = q (q + 1) (q^2 + 1)$ for $n = 4$ and $k = 3$ in my case.

But in $\mathbb{F}_q^n$, in particular for $n=4$, how many points lie on a hyperplane?

Fabius Wiesner
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    As the answer in the second link suggests, the number of points in a $k$ dimensional affine subspace is $q^k$, and it’s independent of $n$ (of course you need $n\ge k$). For a hyperplane, $k=n-1$. – Pranay Dec 12 '24 at 08:52

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