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Codes Cryptogr."],"published-print":{"date-parts":[[2022,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Arguably, the most important open problem in the theory of <jats:italic>q<\/jats:italic>-analogs of designs is the question regarding the existence of a <jats:italic>q<\/jats:italic>-analog <jats:italic>D<\/jats:italic> of the Fano plane. As of today, it remains undecided for every single prime power order <jats:italic>q<\/jats:italic> of the base field. A point <jats:italic>P<\/jats:italic> is called an <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-point of <jats:italic>D<\/jats:italic> if the derived design of <jats:italic>D<\/jats:italic> in <jats:italic>P<\/jats:italic> is a geometric spread. In 1996, Simon Thomas has shown that there always exists a non-<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-point. For the binary case <jats:inline-formula><jats:alternatives><jats:tex-math>$$q = 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-points implies the existence of a partition of the symplectic generalized quadrangle <jats:italic>W<\/jats:italic>(<jats:italic>q<\/jats:italic>) into spreads. As a consequence, the statement of Heden and Sissokho is generalized to all primes <jats:italic>q<\/jats:italic> and all even values of <jats:italic>q<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s10623-022-01033-3","type":"journal-article","created":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T13:07:52Z","timestamp":1651064872000},"page":"1335-1345","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On $$\\alpha $$-points of q-analogs of the Fano plane"],"prefix":"10.1007","volume":"90","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5901-4381","authenticated-orcid":false,"given":"Michael","family":"Kiermaier","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,27]]},"reference":[{"issue":"1","key":"1033_CR1","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/s10623-005-5666-0","volume":"38","author":"S Ball","year":"2006","unstructured":"Ball S., Govaerts P., Storme L.: On ovoids of parabolic quadrics. 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