{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T12:12:00Z","timestamp":1773231120674,"version":"3.50.1"},"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>A small minimal $k$-blocking set $B$ in $\\mathrm{PG}(n,q)$, $q=p^t$, $p$ prime, is a set of less than $3(q^k+1)\/2$ points in $\\mathrm{PG}(n,q)$, such that every $(n-k)$-dimensional space contains at least one point of $B$ and such that no proper subset of $B$ satisfies this property. The linearity conjecture states that all small minimal $k$-blocking sets in $\\mathrm{PG}(n,q)$ are linear over a subfield $\\mathbb{F}_{p^e}$ of $\\mathbb{F}_q$. Apart from a few cases, this conjecture is still open. In this paper, we show that to prove the linearity conjecture for $k$-blocking sets in $\\mathrm{PG}(n,p^t)$, with exponent $e$ and $p^e\\geq 7$, it is sufficient to prove it for one value of $n$ that is at least $2k$. Furthermore, we show that the linearity of small minimal blocking sets in $\\mathrm{PG}(2,q)$ implies the linearity of small minimal $k$-blocking sets in $\\mathrm{PG}(n,p^t)$, with exponent $e$, with $p^e\\geq t\/e+11$.<\/jats:p>","DOI":"10.37236\/446","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:58:39Z","timestamp":1578697119000},"source":"Crossref","is-referenced-by-count":2,"title":["On the Linearity of Higher-Dimensional Blocking Sets"],"prefix":"10.37236","volume":"17","author":[{"given":"G. Van","family":"De Voorde","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,12,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r174\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r174\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T18:18:55Z","timestamp":1579285135000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r174"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,12,10]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/446","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,12,10]]},"article-number":"R174"}}