Hi all,

Suppose that $\mathbf{f}=[f_1, f_2,\ldots,f_m]$ and $\mathbf{g}=[g_1,g_2,\ldots,g_m]$ are two $m$-dimensional vectors. All $f_i$'s are chosen uniformly randomly from a finite field $\mathbb{F}_q$, where $q$ is the finite field size. For convenience, we denote the elements of $\mathbb{F}_q$ as $\{0, 1, 2,\ldots,q-1\}$. Each $g_i$ is distributed as $\mathrm{Pr}(g_i=0)=1-p_i$ and $\mathrm{Pr}(g_i=r)=p_i/(q-1),r=1,2,\ldots,q-1$. Note that $p_i$ might not be equal for each $i\in\{1,2,\ldots,m\}$.

My question is, is that possible to determine the probability that $\mathbf{f}$ and $\mathbf{g}$ are orthogonal, i.e., $\sum_{i=1}^mf_ig_i=0$? If a closed form is not possible, is that possible to bound the probability? What I want to know is, whether such a probability would approach $0$ when I use a sufficiently large $q$. Any suggestions or references are appreciated.

Thanks sincerely.