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Codes Cryptogr."],"published-print":{"date-parts":[[2024,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider 2-designs which admit a group of automorphisms that is flag-transitive and leaves invariant a chain of nontrivial point-partitions. We build on our recent work on 2-designs which are block-transitive but not necessarily flag-transitive. In particular we use the concept of the \u201carray\u201d of a point subset with respect to the chain of point-partitions; the array describes the distribution of the points in the subset among the classes of each partition. We obtain necessary and sufficient conditions on the array in order for the subset to be a block of such a design. By explicit construction we show that for any <jats:inline-formula><jats:alternatives><jats:tex-math>$$s \\ge 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>s<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, there are infinitely many 2-designs admitting a flag-transitive group that preserves an invariant chain of point-partitions of length <jats:italic>s<\/jats:italic>. Moreover an exhaustive computer search, using <jats:sc>Magma<\/jats:sc>, seeking designs with <jats:inline-formula><jats:alternatives><jats:tex-math>$$e_1e_2e_3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>e<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:msub>\n                      <mml:mi>e<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:msub>\n                      <mml:mi>e<\/mml:mi>\n                      <mml:mn>3<\/mml:mn>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> points (where each <jats:inline-formula><jats:alternatives><jats:tex-math>$$e_i\\le 50$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>e<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mn>50<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) and a partition chain of length <jats:inline-formula><jats:alternatives><jats:tex-math>$$s=3$$<\/jats:tex-math><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>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, produced 57 such flag-transitive designs, among which only three designs arise from our construction\u2014so there is still much to learn.\n<\/jats:p>","DOI":"10.1007\/s10623-024-01400-2","type":"journal-article","created":{"date-parts":[[2024,4,20]],"date-time":"2024-04-20T06:02:01Z","timestamp":1713592921000},"page":"2491-2510","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Chain-imprimitive, flag-transitive 2-designs"],"prefix":"10.1007","volume":"92","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4494-4359","authenticated-orcid":false,"given":"Carmen","family":"Amarra","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alice","family":"Devillers","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cheryl E.","family":"Praeger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,4,20]]},"reference":[{"key":"1400_CR1","doi-asserted-by":"crossref","unstructured":"Alavi S.H., Daneshkhah A., Devillers A., Praeger C.E.: Block-transitive two-designs based on grids. 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