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A note on the diameter of graphs of two-dimensional simplex codes
Abstract
Let Γs(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≥5. Together with the elementary cases q=2,3, this gives diamΓs(2,q)=0,2,3 for q=2, q=3, and q≥4, respectively. The upper bound is obtained from a consequence of a theorem of Marshall Hall on finite abelian groups. For q≥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\geq7 we also retain an independent counting proof. The case q=4 is handled separately.
Related
BibTeX
@misc{siemaszko2026note,
title = {{A note on the diameter of graphs of two-dimensional simplex codes}},
author = {Artur Siemaszko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13574},
eprint = {2609.13574},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13574},
}
RIS
TY - UNPB
TI - A note on the diameter of graphs of two-dimensional simplex codes
AU - Artur Siemaszko
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.13574
UR - https://arxiv.org/abs/2609.13574
AB - 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≥5. Together with the elementary cases q=2,3, this gives diamΓ^s(2,q)=0,2,3 for q=2, q=3, and q≥4, respectively. The upper bound is obtained from a consequence of a theorem of Marshall Hall on finite abelian groups. For q≥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\geq7 we also retain an independent counting proof. The case q=4 is handled separately.
ER -