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

Abstract

Let ${\Gamma }^s(2,q)$ 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 $q\ge 5$. Together with the elementary cases $q=2,3$, this gives diam${\Gamma }^s(2,q)=0,2,3$ for $q=2$, $q=3$, and $q\ge 4$, respectively. The upper bound is obtained from a consequence of a theorem of Marshall Hall on finite abelian groups. For $q\ge 5$ an explicit diagonal pair of simplex lines gives the matching lower bound; we give two proofs, one using moments and one using products. For $q\ge 7wealsoretainanindependentcountingproof.Thecase$q=4$ is handled separately.