{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:07Z","timestamp":1753893847429,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>One of the most intriguing problems for $q$-analogs of designs, is the existence question of\u00a0an infinite family of $q$-Steiner systems that are not spreads.\u00a0In particular the most interesting case is the\u00a0existence question for the $q$-analog of the Fano plane, known also as the $q$-Fano plane. These\u00a0questions are in the front line of open problems in block design.\u00a0There was a common belief and a conjecture that such structures do not exist.\u00a0Only recently, $q$-Steiner systems were found for one set of parameters.\u00a0In this paper, a definition for the $q$-analog of the residual design\u00a0is presented. This new definition is different from previous known definition,\u00a0but its properties reflect better the $q$-analog properties.\u00a0The existence of a design with the parameters\u00a0of the residual $q$-Steiner system in general and the\u00a0residual $q$-Fano plane in particular are examined.\u00a0We construct different\u00a0residual $q$-Fano planes for all $q$, where $q$ is a prime power. The constructed structure\u00a0is just one step from a construction of a $q$-Fano plane.<\/jats:p>","DOI":"10.37236\/7107","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:35:14Z","timestamp":1578652514000},"source":"Crossref","is-referenced-by-count":2,"title":["Residual $q$-Fano Planes and Related Structures"],"prefix":"10.37236","volume":"25","author":[{"given":"Tuvi","family":"Etzion","sequence":"first","affiliation":[]},{"given":"Niv","family":"Hooker","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,4,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i2p3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i2p3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:35:05Z","timestamp":1579217705000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i2p3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,3]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,4,13]]}},"URL":"https:\/\/doi.org\/10.37236\/7107","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,4,3]]},"article-number":"P2.3"}}