{"paper":{"title":"Upper bounds on the smallest size of a complete cap in $\\mathrm{PG}(N,q)$, $N\\ge3$, under a certain probabilistic conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Alexander A. Davydov, Fernanda Pambianco, Giorgio Faina, Stefano Marcugini","submitted_at":"2017-06-06T19:42:19Z","abstract_excerpt":"In the projective space $\\mathrm{PG}(N,q)$ over the Galois field of order $q$, $N\\ge3$, an iterative step-by-step construction of complete caps by adding a new point on every step is considered. It is proved that uncovered points are evenly placed on the space. A natural conjecture on an estimate of the number of new covered points on every step is done. For a part of the iterative process, this estimate is proved rigorously. Under the conjecture mentioned, new upper bounds on the smallest size $t_{2}(N,q)$ of a complete cap in $\\mathrm{PG}(N,q)$ are obtained, in particular, \\begin{align*} t_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.01941","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"}