{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:22:38Z","timestamp":1740122558317,"version":"3.37.3"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"11","license":[{"start":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T00:00:00Z","timestamp":1719446400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T00:00:00Z","timestamp":1719446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100007195","name":"Universit\u00e0 degli Studi di Napoli Federico II","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100007195","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2024,11]]},"abstract":"Abstract<\/jats:title>An affine spread<\/jats:italic> is a set of subspaces of $$\\textrm{AG}(n, q)$$<\/jats:tex-math>\n \n AG<\/mml:mtext>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of the same dimension that partitions the points of $$\\textrm{AG}(n, q)$$<\/jats:tex-math>\n \n AG<\/mml:mtext>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Equivalently, an affine spread<\/jats:italic> is a set of projective subspaces of $$\\textrm{PG}(n, q)$$<\/jats:tex-math>\n \n PG<\/mml:mtext>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of the same dimension which partitions the points of $$\\textrm{PG}(n, q) \\setminus H_{\\infty }$$<\/jats:tex-math>\n \n PG<\/mml:mtext>\n \n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\<\/mml:mo>\n \n H<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>; here $$H_{\\infty }$$<\/jats:tex-math>\n \n H<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> denotes the hyperplane at infinity of the projective closure of $$\\textrm{AG}(n, q)$$<\/jats:tex-math>\n \n AG<\/mml:mtext>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Let $$\\mathcal {Q}$$<\/jats:tex-math>\n Q<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be a non-degenerate quadric of $$H_\\infty $$<\/jats:tex-math>\n \n H<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and let $$\\Pi $$<\/jats:tex-math>\n \u03a0<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be a generator of $$\\mathcal {Q}$$<\/jats:tex-math>\n Q<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where $$\\Pi $$<\/jats:tex-math>\n \u03a0<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is a t<\/jats:italic>-dimensional projective subspace. An affine spread $$\\mathcal {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> consisting of $$(t+1)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n t<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-dimensional projective subspaces of $$\\textrm{PG}(n, q)$$<\/jats:tex-math>\n \n PG<\/mml:mtext>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is called hyperbolic, parabolic<\/jats:italic> or elliptic<\/jats:italic> (according as $$\\mathcal {Q}$$<\/jats:tex-math>\n Q<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is hyperbolic, parabolic or elliptic) if the following hold:\n \n Each member of $$\\mathcal {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> meets $$H_\\infty $$<\/jats:tex-math>\n \n H<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in a distinct generator of $$\\mathcal {Q}$$<\/jats:tex-math>\n Q<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> disjoint from $$\\Pi $$<\/jats:tex-math>\n \u03a0<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>;<\/jats:p>\n <\/jats:list-item>\n \n Elements of $$\\mathcal {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> have at most one point in common;<\/jats:p>\n <\/jats:list-item>\n \n If $$S, T \\in \\mathcal {P}$$<\/jats:tex-math>\n \n S<\/mml:mi>\n ,<\/mml:mo>\n T<\/mml:mi>\n \u2208<\/mml:mo>\n P<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $$|S \\cap T| = 1$$<\/jats:tex-math>\n \n |<\/mml:mo>\n S<\/mml:mi>\n \u2229<\/mml:mo>\n T<\/mml:mi>\n |<\/mml:mo>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then $$\\langle S, T \\rangle \\cap \\mathcal {Q}$$<\/jats:tex-math>\n \n \u27e8<\/mml:mo>\n S<\/mml:mi>\n ,<\/mml:mo>\n T<\/mml:mi>\n \u27e9<\/mml:mo>\n \u2229<\/mml:mo>\n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is a hyperbolic quadric of $$\\mathcal {Q}$$<\/jats:tex-math>\n Q<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>\n <\/jats:list-item>\n <\/jats:list> In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of $$\\textrm{PG}(n, q)$$<\/jats:tex-math>\n \n PG<\/mml:mtext>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is equivalent to a spread of $$\\mathcal {Q}^+(n+1, q)$$<\/jats:tex-math>\n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n +<\/mml:mo>\n <\/mml:msup>\n \n (<\/mml:mo>\n n<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $$\\mathcal {Q}(n+1, q)$$<\/jats:tex-math>\n \n Q<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> or $$\\mathcal {Q}^-(n+1, q)$$<\/jats:tex-math>\n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n -<\/mml:mo>\n <\/mml:msup>\n \n (<\/mml:mo>\n n<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, respectively.<\/jats:p>","DOI":"10.1007\/s10623-024-01447-1","type":"journal-article","created":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T10:02:15Z","timestamp":1719482535000},"page":"3495-3502","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Affine vector space partitions and spreads of quadrics"],"prefix":"10.1007","volume":"92","author":[{"given":"Somi","family":"Gupta","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesco","family":"Pavese","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,6,27]]},"reference":[{"key":"1447_CR1","doi-asserted-by":"publisher","first-page":"156","DOI":"10.1007\/BF01187370","volume":"60","author":"J Andr\u00e9","year":"1954","unstructured":"Andr\u00e9 J.: \u00dcber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. 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