{"paper":{"title":"On Almost Complete Subsets of a Conic in $\\mathrm{PG}(2,q)$, Completeness of Normal Rational Curves and Extendability of Reed-Solomon Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander A. Davydov, Daniele Bartoli, Fernanda Pambianco, Stefano Marcugini","submitted_at":"2016-09-19T10:29:53Z","abstract_excerpt":"A subset $\\mathcal{S}$ of a conic $\\mathcal{C}$ in the projective plane $\\mathrm{PG}(2,q)$ is called almost complete (AC-subset for short) if it can be extended to a larger arc in $\\mathrm{PG}(2,q)$ only by the points of $\\mathcal{C}\\setminus \\mathcal{S}$ and by the nucleus of $\\mathcal{C}$ when $q$ is even. New upper bounds on the smallest size $t(q)$ of an AC-subset are obtained, in particular, \\begin{align*} &t(q)<\\sqrt{q(3\\ln q+\\ln\\ln q +\\ln3)}+\\sqrt{\\frac{q}{3\\ln q}}+4\\thicksim\\sqrt{3q\\ln q};&t(q)<1.835\\sqrt{q\\ln q}.\\end{align*} The new bounds are used to increase regions of pairs $(N,q)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05657","kind":"arxiv","version":3},"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"}