The crossing number and the unit-distance crossing number of the Hamming graphs H(d,3)=K_3^{box d}, and their realizations over many coordinate fields

The Hamming graph H(d,3)=K_3^{box d} (n=3^d vertices) is the graph of single-symbol errors of ternary codes, yet we ask a geometric question of it: can it be drawn in the plane with every edge exactly one unit long and no two non-adjacent vertices a unit apart? It can, for every d, joining two probl