Year: 2026
Type: preprint
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 $q\geq5$. Together with the elementary cases $q=2,3$, this gives diam$Γ^s(2,q)=0,2,3$ for $q=2$, $q=3$, and $q\geq4$, respectively. The upper bound is obtained from a consequence of a theorem of Marshall Hall on finite abelian groups. For $q\geq5$ 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\geq7 we also retain an independent counting proof. The case $q=4$ is handled separately.
Source: arXiv (Cornell University)
Author: Siemaszko, Artur
Institution: University of Warmia and Mazury in Olsztyn
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213297975