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Codes Cryptogr."],"published-print":{"date-parts":[[2022,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>If an <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {F}}_q$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mi>q<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-linear set <jats:inline-formula><jats:alternatives><jats:tex-math>$$L_U$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mi>U<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in a projective space is defined by a vector subspace <jats:italic>U<\/jats:italic> which is linear over a proper superfield of <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {F}}_{q}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mi>q<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then all of its points have weight at least 2. It is known that the converse of this statement holds for linear sets of rank <jats:italic>h<\/jats:italic> in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathrm {PG}(1,q^h)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>PG<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>q<\/mml:mi>\n                      <mml:mi>h<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> but for linear sets of rank <jats:inline-formula><jats:alternatives><jats:tex-math>$$k&lt;h$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>&lt;<\/mml:mo>\n                    <mml:mi>h<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> the converse of this statement is in general no longer true. The first part of this paper studies the relation between the weights of points and the size of a linear set, and introduces the concept of the <jats:italic>geometric field of linearity<\/jats:italic> of a linear set. This notion will allow us to show the main theorem, stating that for particular linear sets without points of weight 1, the converse of the above statement still holds as long as we take the geometric field of linearity into account.<\/jats:p>","DOI":"10.1007\/s10623-022-01011-9","type":"journal-article","created":{"date-parts":[[2022,2,3]],"date-time":"2022-02-03T14:04:15Z","timestamp":1643897055000},"page":"779-799","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The geometric field of linearity of linear sets"],"prefix":"10.1007","volume":"90","author":[{"given":"Dibyayoti","family":"Jena","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4957-6911","authenticated-orcid":false,"given":"Geertrui","family":"Van de Voorde","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,2,3]]},"reference":[{"key":"1011_CR1","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1016\/j.jcta.2003.09.006","volume":"104","author":"S Ball","year":"2003","unstructured":"Ball S.: The number of directions determined by a function over a finite field. 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