{"paper":{"title":"A coding theoretic approach to the uniqueness conjecture for projective planes of prime order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bhaskar Bagchi","submitted_at":"2018-01-22T11:02:44Z","abstract_excerpt":"An outstanding folklore conjecture asserts that, for any prime $p$, up to isomorphism the projective plane $PG(2,\\mathbb{F}_p)$ over the field $\\mathbb{F}_p := \\mathbb{Z}/p\\mathbb{Z}$ is the unique projective plane of order $p$. Let $\\pi$ be any projective plane of order $p$. For any partial linear space ${\\cal X}$, define the inclusion number $i({\\cal X},\\pi)$ to be the number of isomorphic copies of ${\\cal X}$ in $\\pi$. In this paper we prove that if ${\\cal X}$ has at most $\\log_2 p$ lines, then $i({\\cal X},\\pi)$ can be written as an explicit rational linear combination (depending only on ${"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.07038","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}