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

Abstract

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 $q\ge5$. Together with the elementary cases $q=2,3$, this gives $diamΓ^s(2,q)=0,2,3$ for $q=2$, $q=3$, and $q\ge4$, respectively. The upper bound is obtained from a consequence of a theorem of Marshall Hall on finite abelian groups. For $q\ge7$ a counting argument gives the matching lower bound, while $q=5$ is settled by an explicit pair of simplex lines and a moment obstruction. We also record a short alternative proof for $q=4$ and an independent product argument for $q=5$.