Number of vectors orthogonal to a fixed vector

Asked Nov 04 '17 at 15:51

Active Nov 04 '17 at 16:11

Viewed 62 times

I have a $k$-sized vector $(x_1,x_2, \ldots, x_k)$ where each $x_i$ has been drawn randomly from $\mathbb{Z}/n\mathbb{Z}$.

What is the average number of vectors ($\in \mathbb{Z}^k_n$) orthogonal to it?
If I have such a concrete vector, with $k=4$ and $n=5$, for instance $(2,0,4,1)$, How can I compute the number of orthogonal vectors to it?

Thank you.

edited Nov 04 '17 at 16:11

asked Nov 04 '17 at 15:51

Dingo13

@copper.hat Each $x_i$ is picked (uniformly) at random in $\mathbb{Z}_n$. – Dingo13 Nov 04 '17 at 15:54
I realised that the $n$ was a subscript after my comment, Excuse the noise. – copper.hat Nov 04 '17 at 15:54
$\mathbb{Z}_n$ is the integers modulo $n$? – Wouter Nov 04 '17 at 15:56
@Wouter Yes, it's $\mathbb{Z}/n\mathbb{Z}$. I've updated my post. – Dingo13 Nov 04 '17 at 15:58
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Maybe related: https://math.stackexchange.com/questions/621117/orthogonal-complement-in-a-finite-field-mathbb-zn-q – N74 Nov 04 '17 at 16:34
Just curious, do you mean orthogonal in the $\mathbb{R}$ sense or in a $\mathbb{Z}/n\mathbb{Z}$ sense? – copper.hat Nov 05 '17 at 18:16

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