{"paper":{"title":"Smallest cyclically covering subspaces of $\\mathbb{F}_q^n$, and lower bounds in Isbell's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"David Ellis, Peter Cameron, William Raynaud","submitted_at":"2018-10-08T14:19:11Z","abstract_excerpt":"For a prime power $q$ and a positive integer $n$, we say a subspace $U$ of ${\\mathbb{F}_q^n}$ is {\\em cyclically covering} if the union of the cyclic shifts of $U$ is equal to $\\mathbb{F}_q^n$. We investigate the problem of determining the minimum possible dimension of a cyclically covering subspace of $\\mathbb{F}_q^n$. (This is a natural generalisation of a problem posed in 1991 by the first author.) We prove several upper and lower bounds, and for each fixed $q$, we answer the question completely for infinitely many values of $n$ (which take the form of certain geometric series). Our results"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.03485","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"}