**Definition.** Let $G = (V;E)$ be a finite, undirected graph. $R = \{r_1, \ldots, r_k \} \subseteq V$. *$R$ resolves $G$* if
$$
V \to [0, \infty]^k, v \mapsto (d_G(v,r_1), \ldots, d_G(v, r_k))
$$
is injective (where $d_G$ is the graph metric associated to $G$).

The *metric dimension of $G$* is the cardinality of the smallest resolving $R \subseteq V$.

**Definition.** Let $d,q \in \mathbb N$. The *Hamming graph $H(d,q)$* is the graph with vertex set $\{1, \ldots, q\}^d$ in which two vertices $u = (u_1, \ldots, u_d), v = (v_1, \ldots, v_d)$ are adjacent iff they differ in a single spot, i.e. $| \{ i \mid v_i \neq u_i \} | = 1$.

It is known that the metric dimension of $H(d,q)$ is $\frac{(2 + o(1)) \cdot d}{\log_q(d)}$ (1).

I'm interested in explicit (2) resolving sets for Hamming graphs in the region of $d \sim 1000$ that come reasonably close (3) to their dimension metrics.

Unfortunately, since this lies outside of my area of expertise and I wasn't able to come up with an answer in the literature, I don't know whether this is even remotely possible.

(1) Since I'm unfamiliar with some of the notation in the published literature, I'm not 100% sure about this.

(2) as in computable with current day technology -- ideally already known and publicly available

(3) i.e. significantly smaller than $d$ as $\{ (\delta_{i,j})_{i = 1, \ldots, d} \mid j = 1, \ldots, d \}$ is a trivial resolving set of size $d$