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Codes Cryptogr."],"published-print":{"date-parts":[[2025,2]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>In this paper, we give a geometric construction of the three strong non-lifted <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$(3\\mod 5)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                    <mml:mspace\/>\n                    <mml:mo>mod<\/mml:mo>\n                    <mml:mspace\/>\n                    <mml:mn>5<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-arcs in <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${{\\,\\textrm{PG}\\,}}(3,5)$$<\/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:mn>3<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>5<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> of respective sizes 128, 143, and 168, and construct an infinite family of non-lifted, strong <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$(t\\mod q)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mspace\/>\n                    <mml:mo>mod<\/mml:mo>\n                    <mml:mspace\/>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-arcs in <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${{\\,\\textrm{PG}\\,}}(r,q)$$<\/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>r<\/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> with <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$t=(q+1)\/2$$<\/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:mo>(<\/mml:mo>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> for all <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$r\\ge 3$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>r<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and all odd prime powers <jats:italic>q<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s10623-023-01290-w","type":"journal-article","created":{"date-parts":[[2023,8,27]],"date-time":"2023-08-27T13:01:56Z","timestamp":1693141316000},"page":"359-372","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The geometry of $$(t\\mod 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