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

Let $Γ^s(2,q)$ be the graph induced in the Grassmann graph by the $q$-ary simplex codes of dimension 2. For $q=4$, this graph is known to have diameter 3. We prove that the same diameter occurs for every prime power $q\geq5$. Together with the elementary cases $q=2,3$, this gives diam$Γ^s(2,q)=0,2,3