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_2(d)}$ (1).
I’m interested in explicit (2) resolving sets for Hamming graphs in the region of $ d \sim 1000$ that come reasonably close 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
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