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Codes Cryptogr."],"published-print":{"date-parts":[[2025,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>In this paper, we study the cardinality of the smallest set of lines of the finite projective spaces <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${{\\,\\textrm{PG}\\,}}(n, q)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace\/>\n                      <mml:mtext>PG<\/mml:mtext>\n                      <mml:mspace\/>\n                    <\/mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> such that every plane is incident with at least one line of the set. This is the first main open problem concerning the minimum size of (<jats:italic>s<\/jats:italic>,\u00a0<jats:italic>t<\/jats:italic>)-blocking sets in <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${{\\,\\textrm{PG}\\,}}(n,q)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace\/>\n                      <mml:mtext>PG<\/mml:mtext>\n                      <mml:mspace\/>\n                    <\/mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, where we set <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$s=2$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>s<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$t=1$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. In <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${{\\,\\textrm{PG}\\,}}(n,q)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace\/>\n                      <mml:mtext>PG<\/mml:mtext>\n                      <mml:mspace\/>\n                    <\/mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, an (<jats:italic>s<\/jats:italic>,\u00a0<jats:italic>t<\/jats:italic>)-blocking set refers to a set of <jats:italic>t<\/jats:italic>-spaces such that each <jats:italic>s<\/jats:italic>-space is incident with at least one chosen <jats:italic>t<\/jats:italic>-space. This is a notoriously difficult problem, as it is equivalent to determining the size of certain <jats:italic>q<\/jats:italic>-Tur\u00e1n designs and <jats:italic>q<\/jats:italic>-covering designs. We present an improvement on the upper bounds of Etzion and of Metsch via a refined scheme for a recursive construction, which in fact enables improvement in the general case as well.\n<\/jats:p>","DOI":"10.1007\/s10623-025-01678-w","type":"journal-article","created":{"date-parts":[[2025,7,6]],"date-time":"2025-07-06T08:03:18Z","timestamp":1751788998000},"page":"4403-4432","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Blocking planes by lines in $${{\\,\\textrm{PG}\\,}}(n,q)$$"],"prefix":"10.1007","volume":"93","author":[{"given":"Benedek","family":"Kov\u00e1cs","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zolt\u00e1n L\u00f3r\u00e1nt","family":"Nagy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"D\u00e1vid R.","family":"Szab\u00f3","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,7,6]]},"reference":[{"key":"1678_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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