{"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."}],"OA_type":"green","language":[{"iso":"eng"}],"OA_place":"repository","day":"01","oa_version":"Preprint","_id":"22202","doi":"10.1002/jcd.21522","publication_status":"published","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1409.0148","open_access":"1"}],"page":"393-405","article_processing_charge":"No","article_type":"original","issue":"9","status":"public","date_created":"2026-06-29T13:00:27Z","title":"Hadamard matrices modulo p and small modular Hadamard matrices","publisher":"Wiley","arxiv":1,"type":"journal_article","year":"2016","date_updated":"2026-07-14T11:35:58Z","external_id":{"arxiv":["1409.0148"]},"citation":{"short":"V.Z. Kuperberg, Journal of Combinatorial Designs 24 (2016) 393–405.","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>","ista":"Kuperberg VZ. 2016. Hadamard matrices modulo p and small modular Hadamard matrices. Journal of Combinatorial Designs. 24(9), 393–405.","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.","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>.","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>."},"scopus_import":"1","keyword":["modular hadamard matrices","modular symmetric designs"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","month":"09","extern":"1","date_published":"2016-09-01T00:00:00Z","volume":24,"intvolume":"        24","author":[{"first_name":"Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697","last_name":"Kuperberg","full_name":"Kuperberg, Vivian Zieve"}],"publication_identifier":{"eissn":["1520-6610"],"issn":["1063-8539"]},"publication":"Journal of Combinatorial Designs"}