How large a subset of $\mathbb{F}_q^d$ can determine all determinants? Denote by $\mathbb{F}_q$ a finite field with $q$ elements. For $\mathcal{P}$ be arbitrary subset of $\mathbb{F}_q^d.$ We define the set 
$$S:= \left\{ \det([x_1,x_2,\dots,x_d]): x_1,x_2,\dots,x_d \in \mathcal{P}\right\},$$ 
where $[x_1,x_2,\dots,x_d]$ is $d \times d$ matrix with $d$ column vectors $x_1,x_2,\dots,x_d.$ 

What is the largest cardinality of  $\mathcal{P}$ such that $S \neq \mathbb{F}_q$?

(equivalently, What is the minimum value of $|\mathcal{P}|$ to make sure that $S=\mathbb{F}_q?$)
Comment: If we consider a subset $\mathcal{Q}$ of $M_d(\mathbb{F}_q)$ and 
$$S:=\left\{\det(A): A \in \mathcal{Q}\right\}.$$
Then, the minimum of cardinality of $\mathcal{Q}$ such that $S=\mathbb{F}_q$ for sure is 
$$N = q^{d^2} - \dfrac{|\mathrm{GL}_d(\mathbb{F}_q)|}{q-1} + 1.$$
How about the previous question? 
 A: Based on a conversation with Lspice, here is an idea to play with to (for large q) get away with much fewer than q+d vectors. I will show a non optimal example.
Let d be at least 4, and q sufficiently large (bigger than 1000). I pick q=1009. The first row will have ten powers of two, from 1 to 512. The second will have seven powers of three, from 1 to 729. The third row will have powers of five from 1 to 625. (Each of these are really the power times the ith basis vector). The last row will have the remaining primes, and where needed, products of smaller primes. This will be 165 primes, about a similar number of products of two primes, and a smattering (less than four hundred all told) of products of three primes, not necessarily distinct. This gives less than 500 vectors to give positive determinants up to 1008. 
Of course one needs d distinct vectors to give a nonzero determinant, but not much more to get all values of the field as determinants. And this is computing determinants as if they were integer matrices. Further savings can be realized if one does the determinant computation over the field itself.
I do not have a good sense of asymptotic or even a good lower bound for this. I imagine one needs a quantity like k(q^(1/k)) involved, but I can't show that is a lower bound.
Gerhard "Has Got This Party Started" Paseman, 2019.11.26.
