[{"scopus_import":"1","acknowledgement":"We are very grateful to Matthew Kwan and Alp Müyesser with whom we had many interesting discussions leading to the results of this note. We also thank the anonymous reviewers for their suggestions improving the presentation of this note.\r\n\r\nMA was supported by the Austrian Science Fund (FWF) [10.55776/ESP3863424] and by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant—project number 101034413. PM was supported by the European Union's Horizon Europe Marie Skłodowska-Curie grant RAND-COMB-DESIGN—project number 101106032.","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"09","publisher":"Wiley","arxiv":1,"type":"journal_article","year":"2025","citation":{"chicago":"Anastos, Michael, and Patrick Morris. “A Note on Finding Large Transversals Efficiently.” <i>Journal of Combinatorial Designs</i>. Wiley, 2025. <a href=\"https://doi.org/10.1002/jcd.21990\">https://doi.org/10.1002/jcd.21990</a>.","ieee":"M. Anastos and P. Morris, “A note on finding large transversals efficiently,” <i>Journal of Combinatorial Designs</i>, vol. 33, no. 9. Wiley, pp. 338–342, 2025.","ista":"Anastos M, Morris P. 2025. A note on finding large transversals efficiently. Journal of Combinatorial Designs. 33(9), 338–342.","mla":"Anastos, Michael, and Patrick Morris. “A Note on Finding Large Transversals Efficiently.” <i>Journal of Combinatorial Designs</i>, vol. 33, no. 9, Wiley, 2025, pp. 338–42, doi:<a href=\"https://doi.org/10.1002/jcd.21990\">10.1002/jcd.21990</a>.","ama":"Anastos M, Morris P. A note on finding large transversals efficiently. <i>Journal of Combinatorial Designs</i>. 2025;33(9):338-342. doi:<a href=\"https://doi.org/10.1002/jcd.21990\">10.1002/jcd.21990</a>","short":"M. Anastos, P. Morris, Journal of Combinatorial Designs 33 (2025) 338–342.","apa":"Anastos, M., &#38; Morris, P. (2025). A note on finding large transversals efficiently. <i>Journal of Combinatorial Designs</i>. Wiley. <a href=\"https://doi.org/10.1002/jcd.21990\">https://doi.org/10.1002/jcd.21990</a>"},"external_id":{"arxiv":["2412.05891"],"isi":["001495472300001"]},"date_updated":"2025-12-30T08:37:37Z","project":[{"_id":"8f906bd2-16d5-11f0-9cad-e07be8aa9ac9","name":"Combinatorial Optimisation Problems on Sparse Random Graphs","grant_number":"ESP3863424"},{"grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020"}],"publication_identifier":{"eissn":["1520-6610"],"issn":["1063-8539"]},"publication":"Journal of Combinatorial Designs","date_published":"2025-09-01T00:00:00Z","volume":33,"intvolume":"        33","author":[{"full_name":"Anastos, Michael","last_name":"Anastos","id":"0b2a4358-bb35-11ec-b7b9-e3279b593dbb","first_name":"Michael"},{"first_name":"Patrick","last_name":"Morris","full_name":"Morris, Patrick"}],"doi":"10.1002/jcd.21990","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2412.05891","open_access":"1"}],"publication_status":"published","isi":1,"abstract":[{"lang":"eng","text":"In an  n×n  array filled with symbols, a transversal is a collection of entries with distinct rows, columns and symbols. In this note we show that if no symbol appears more than  βn  times, the array contains a transversal of size  (1−β/4−o(1))n . In particular, if the array is filled with  n  symbols, each appearing  n  times (an equi- n  square), we get transversals of size  (3/4−o(1))n. Moreover, our proof gives a deterministic algorithm with polynomial running time, that finds these transversals."}],"oa":1,"OA_type":"green","day":"01","OA_place":"repository","language":[{"iso":"eng"}],"oa_version":"Preprint","_id":"19798","issue":"9","department":[{"_id":"MaKw"}],"status":"public","date_created":"2025-06-08T22:01:23Z","title":"A note on finding large transversals efficiently","ec_funded":1,"page":"338-342","article_processing_charge":"No","article_type":"original"},{"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1409.0148","open_access":"1"}],"publication_status":"published","doi":"10.1002/jcd.21522","_id":"22202","OA_place":"repository","language":[{"iso":"eng"}],"day":"01","oa_version":"Preprint","OA_type":"green","oa":1,"abstract":[{"lang":"eng","text":"We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the 7-modular and 11-modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a conjectural sufficient condition for the existence of a p-modular Hadamard matrix for all but finitely many cases. When 2 is a primitive root of a prime p, we conditionally solve this conjecture and therefore the p-modular version of the Hadamard conjecture for all but finitely many cases when p ≡ 3(mod 4), and prove a weaker result for p ≡ 1 (mod 4). Finally, we look at constraints on the existence of m-modular Hadamard matrices when the size of the matrix is small compared to m."}],"issue":"9","status":"public","date_created":"2026-06-29T13:00:27Z","title":"Hadamard matrices modulo p and small modular Hadamard matrices","article_type":"original","article_processing_charge":"No","page":"393-405","extern":"1","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"09","keyword":["modular hadamard matrices","modular symmetric designs"],"scopus_import":"1","citation":{"apa":"Kuperberg, V. Z. (2016). Hadamard matrices modulo p and small modular Hadamard matrices. <i>Journal of Combinatorial Designs</i>. Wiley. <a href=\"https://doi.org/10.1002/jcd.21522\">https://doi.org/10.1002/jcd.21522</a>","short":"V.Z. Kuperberg, Journal of Combinatorial Designs 24 (2016) 393–405.","ama":"Kuperberg VZ. Hadamard matrices modulo p and small modular Hadamard matrices. <i>Journal of Combinatorial Designs</i>. 2016;24(9):393-405. doi:<a href=\"https://doi.org/10.1002/jcd.21522\">10.1002/jcd.21522</a>","mla":"Kuperberg, Vivian Zieve. “Hadamard Matrices modulo p and Small Modular Hadamard Matrices.” <i>Journal of Combinatorial Designs</i>, vol. 24, no. 9, Wiley, 2016, pp. 393–405, doi:<a href=\"https://doi.org/10.1002/jcd.21522\">10.1002/jcd.21522</a>.","ieee":"V. Z. Kuperberg, “Hadamard matrices modulo p and small modular Hadamard matrices,” <i>Journal of Combinatorial Designs</i>, vol. 24, no. 9. Wiley, pp. 393–405, 2016.","ista":"Kuperberg VZ. 2016. Hadamard matrices modulo p and small modular Hadamard matrices. Journal of Combinatorial Designs. 24(9), 393–405.","chicago":"Kuperberg, Vivian Zieve. “Hadamard Matrices modulo p and Small Modular Hadamard Matrices.” <i>Journal of Combinatorial Designs</i>. Wiley, 2016. <a href=\"https://doi.org/10.1002/jcd.21522\">https://doi.org/10.1002/jcd.21522</a>."},"date_updated":"2026-07-14T11:35:58Z","external_id":{"arxiv":["1409.0148"]},"type":"journal_article","year":"2016","publisher":"Wiley","arxiv":1,"publication":"Journal of Combinatorial Designs","publication_identifier":{"issn":["1063-8539"],"eissn":["1520-6610"]},"author":[{"last_name":"Kuperberg","full_name":"Kuperberg, Vivian Zieve","first_name":"Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697"}],"volume":24,"intvolume":"        24","date_published":"2016-09-01T00:00:00Z"}]