{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:37Z","timestamp":1753893817271,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>\u00a0 A $[t]$-trade is a pair $T=(T_+, T_-)$ of disjoint collections of subsets (blocks) of a $v$-set $V$ such that for every $0\\le i\\le t$, any $i$-subset of $V$ is included in the same number of blocks of $T_+$ and of $T_-$. It follows that $|T_+| = |T_-|$ and this common value is called the volume of $T$. If we restrict all the blocks to have the same size, we obtain the\u00a0 classical $t$-trades as a special case of $[t]$-trades. It is known that the minimum volume of a nonempty $[t]$-trade is $2^t$. Simple $[t]$-trades (i.e., those with no repeated blocks) correspond to a Boolean function of degree at most $v-t-1$. From the characterization of Kasami\u2013Tokura of such functions with small number of ones, it is known that\u00a0 any simple $[t]$-trade of volume at most $2\\cdot2^t$ belongs to one of two affine types, called Type (A) and Type (B) where Type (A) $[t]$-trades are known to exist. By considering the affine rank, we prove that $[t]$-trades of Type (B) do not exist. Further, we derive the spectrum of volumes of simple trades up to $2.5\\cdot 2^t$, extending the known result for volumes less than $2\\cdot 2^t$. We also give a characterization of \"small\" $[t]$-trades for $t=1,2$. Finally, an algorithm to produce $[t]$-trades for specified\u00a0 $t$, $v$ is given. The result of the implementation of the algorithm for $t\\le4$, $v\\le7$ is reported.<\/jats:p>","DOI":"10.37236\/8367","type":"journal-article","created":{"date-parts":[[2020,1,24]],"date-time":"2020-01-24T09:18:37Z","timestamp":1579857517000},"source":"Crossref","is-referenced-by-count":1,"title":["On the Volumes and Affine Types of Trades"],"prefix":"10.37236","volume":"27","author":[{"given":"Ebrahim","family":"Ghorbani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sara","family":"Kamali","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gholamreza B.","family":"Khosrovshahi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Denis","family":"Krotov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,1,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p29\/8016","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p29\/8016","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,24]],"date-time":"2020-01-24T09:18:37Z","timestamp":1579857517000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i1p29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,24]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/8367","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,1,24]]},"article-number":"P1.29"}}