A note on the diameter of graphs of two-dimensional simplex codes

Let $\text{Gs}(2,q)$ be the subgraph of the Grassmann graph on the $2$-dimensional subspaces of ${\text{F}}_q^{q+1}$ induced by the $q$-ary simplex codes of dimension $2$. The cases $q=2,3$ are elementary, while Kwiatkowski and Pankov determined the complete distance relation for $q=4$ and, in particular, proved that $\text{diam}\text{Gs}(2,4)=3$. We prove that $$\text{diam}\text{Gs}(2,q)=3\text{for every prime power }q\ge 5.$$ Consequently the diameter of $\text{Gs}(2,q)$ is completely determined for all $q$: it is $0$ for $q=2$, $2$ for $q=3$, and $3$ for $q\ge 4$. The main ingredient is a bridge lemma based on a theorem of Marshall Hall on finite abelian groups; it yields the uniform upper bound $3$. For $q\ge 7$ the lower bound follows from a counting argument. The remaining cases are treated by short moment calculations; for $q=5$ we also give an independent product proof.