Let $k$ be a field with five elements. Let $V$ be the $k$-vector space of $5\times1$ matrices with entries in $k$. Let $S$ be a subset of $V$ such that $u^t v=0$ for all $u$ and $v$ in $S$. What is the maximum possible cardinality of $S$ ?
The question seems to be asking the maximum number of mutually orthogonal vectors over $k$. How do we compute it? Is it same as the number of orthogonal matrices over $k$? Thanks beforehand.
Since any linear combination of vectors that fulfil the condition for the elements of $S$ also fulfils the condition, a maximal $S$ is a subspace of $V$. This subspace is self-orthogonal. That is, if its dimension is $k$, its elements fulfill $k$ linearly independent linear equations imposed by the orthogonality. This can only be the case if $k\le5-k$ and thus $k\le2$. We can exhibit a basis for a subspace of dimension $2$ that fulfils the condition: $(1,2,0,0,0)$, $(0,0,1,2,0)$. Thus the maximal dimension $k$ is $2$, and a maximal $S$ has $5^2=25$ elements.