{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T05:04:32Z","timestamp":1761800672028,"version":"build-2065373602"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"11","license":[{"start":{"date-parts":[[2025,7,26]],"date-time":"2025-07-26T00:00:00Z","timestamp":1753488000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,26]],"date-time":"2025-07-26T00:00:00Z","timestamp":1753488000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP200100080","DP200100080"],"award-info":[{"award-number":["DP200100080","DP200100080"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001801","name":"University of Western Australia","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001801","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":[[2025,11]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    It was shown in 1989 by Delandtsheer and Doyen that, for a 2-design with\n                    <jats:italic>v<\/jats:italic>\n                    points and block size\n                    <jats:italic>k<\/jats:italic>\n                    , a block-transitive group of automorphisms can be point-imprimitive (that is, leave invariant a nontrivial partition of the point set) only if\n                    <jats:italic>v<\/jats:italic>\n                    is small enough relative to\n                    <jats:italic>k<\/jats:italic>\n                    . Recently, exploiting a construction of block-transitive point-imprimitive 2-designs given by Cameron and the last author, four of the authors studied 2-designs admitting a block-transitive group that preserves a two-dimensional grid structure on the point set. Here we consider the case where there a block-transitive group preserves a multidimensional grid structure on points. We provide necessary and sufficient conditions for such 2-designs to exist in terms of the parameters of the grid, and certain \u2018array parameters\u2019 which describe a subset of points (which will be a block of the design). Using this criterion, we construct explicit examples of 2-designs for grids of dimensions three and four, and pose several open questions.\n                  <\/jats:p>","DOI":"10.1007\/s10623-025-01700-1","type":"journal-article","created":{"date-parts":[[2025,7,26]],"date-time":"2025-07-26T14:27:11Z","timestamp":1753540031000},"page":"4683-4719","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Higher-dimensional grid-imprimitive block-transitive designs"],"prefix":"10.1007","volume":"93","author":[{"given":"Seyed Hassan","family":"Alavi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carmen","family":"Amarra","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ashraf","family":"Daneshkhah","sequence":"additional","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":[[2025,7,26]]},"reference":[{"key":"1700_CR1","doi-asserted-by":"crossref","unstructured":"Aigner M.: Combinatorial Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 234. Springer, Berlin (1979).","DOI":"10.1007\/978-1-4615-6666-3"},{"key":"1700_CR2","doi-asserted-by":"publisher","first-page":"592","DOI":"10.1112\/blms.12744","volume":"55","author":"SH Alavi","year":"2023","unstructured":"Alavi S.H., Daneshkhah A., Devillers A., Praeger C.E.: Block-transitive two-designs based on grids. Bull. Lond. Math. Soc. 55, 592\u2013610 (2023). https:\/\/doi.org\/10.1112\/blms.12744.","journal-title":"Bull. Lond. Math. Soc."},{"key":"1700_CR3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2024.105866","volume":"205","author":"C Amarra","year":"2024","unstructured":"Amarra C., Devillers A., Praeger C.E.: Block-transitive $$2$$-designs with a chain of imprimitive partitions. J. Combin. Theory Ser. A 205, 105866 (2024).","journal-title":"J. Combin. Theory Ser. A"},{"key":"1700_CR4","doi-asserted-by":"publisher","first-page":"117","DOI":"10.2140\/iig.2008.8.117","volume":"8","author":"A Betten","year":"2009","unstructured":"Betten A., Cresp G., Praeger C.E.: Line-transitive, point-imprimitive linear spaces: the grid case. Innov. Incidence Geom. 8, 117\u2013135 (2009).","journal-title":"Innov. Incidence Geom."},{"key":"1700_CR5","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/BF01111448","volume":"96","author":"RE Block","year":"1967","unstructured":"Block R.E.: On the orbits of collineation groups. Math. Z. 96, 33\u201349 (1967).","journal-title":"Math. Z."},{"key":"1700_CR6","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","volume":"24","author":"W Bosma","year":"1997","unstructured":"Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235\u2013265 (1997).","journal-title":"J. Symb. 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Mat."},{"key":"1700_CR9","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/0012-365X(93)90051-T","volume":"118","author":"PJ Cameron","year":"1993","unstructured":"Cameron P.J., Praeger C.E.: Block-transitive $$t$$-designs I: point-imprimitive designs. Discret. Math. 118, 33\u201343 (1993).","journal-title":"Discret. Math."},{"key":"1700_CR10","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1007\/BF00572446","volume":"29","author":"A Delandtsheer","year":"1989","unstructured":"Delandtsheer A., Doyen J.: Most block-transitive $$t$$-designs are point-primitive. Geom. Dedicata. 29, 307\u2013310 (1989).","journal-title":"Geom. Dedicata."},{"key":"1700_CR11","first-page":"204","volume":"7","author":"QM Husain","year":"1945","unstructured":"Husain Q.M.: On the totality of the solutions for the symmetrical incomplete block designs: $$\\lambda =2, k=5$$ or $$6$$. Sankhy\u0101 Indian J. Stat. 7, 204\u2013208 (1945).","journal-title":"Sankhy\u0101 Indian J. 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