{"paper":{"title":"Nontrivial t-Designs over Finite Fields Exist for All t","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Vardy, Arman Fazeli, Shachar Lovett","submitted_at":"2013-06-10T02:14:31Z","abstract_excerpt":"A $t$-$(n,k,\\lambda)$ design over $\\F_q$ is a collection of $k$-dimensional subspaces of $\\F_q^n$, called blocks, such that each $t$-dimensional subspace of $\\F_q^n$ is contained in exactly $\\lambda$ blocks. Such $t$-designs over $\\F_q$ are the $q$-analogs of conventional combinatorial designs. Nontrivial $t$-$(n,k,\\lambda)$ designs over $\\F_q$ are currently known to exist only for $t \\leq 3$. Herein, we prove that simple (meaning, without repeated blocks) nontrivial $t$-$(n,k,\\lambda)$ designs over $\\F_q$ exist for all $t$ and $q$, provided that $k > 12t$ and $n$ is sufficiently large. This m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.2088","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"}