{"paper":{"title":"On the gaps of the spectrum of volumes of trades","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Denis S. Krotov (Sobolev Institute of Mathematics, Novosibirsk, Russia)","submitted_at":"2018-10-31T18:00:01Z","abstract_excerpt":"A pair $\\{T_0,T_1\\}$ of disjoint collections of $k$-subsets (blocks) of a set $V$ of cardinality $v$ is called a $t$-$(v,k)$ trade or simply a $t$-trade if every $t$-subset of $V$ is included in the same number of blocks of $T_0$ and $T_1$. The cardinality of $T_0$ is called the volume of the trade. Using the weight distribution of the Reed--Muller code, we prove the conjecture that for every $i$ from $2$ to $t$, there are no $t$-trades of volume greater than $2^{t+1}-2^i$ and less than $2^{t+1}-2^{i-1}$ and derive restrictions on the $t$-trade volumes that are less than $2^{t+1}+2^{t-1}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.00015","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"}