{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T23:05:34Z","timestamp":1768518334871,"version":"3.49.0"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"10","license":[{"start":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T00:00:00Z","timestamp":1717459200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T00:00:00Z","timestamp":1717459200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["2123224"],"award-info":[{"award-number":["2123224"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2024,10]]},"DOI":"10.1007\/s10623-024-01430-w","type":"journal-article","created":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T08:03:06Z","timestamp":1717488186000},"page":"3125-3131","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A note on approximate Hadamard matrices"],"prefix":"10.1007","volume":"92","author":[{"given":"Stefan","family":"Steinerberger","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,6,4]]},"reference":[{"key":"1430_CR1","doi-asserted-by":"crossref","unstructured":"Agaian S.S.: Hadamard Matrices and Their Applications, vol. 1168. Lecture Notes in Mathematics. Springer, Berlin (1985).","DOI":"10.1007\/BFb0101073"},{"key":"1430_CR2","doi-asserted-by":"publisher","first-page":"977","DOI":"10.4007\/annals.2020.192.3.6","volume":"192","author":"P Balister","year":"2020","unstructured":"Balister P., Bollob\u00e1s B., Morris R., Sahasrabudhe J., Tiba M.: Flat Littlewood polynomials exist. Ann. Math. 192, 977\u20131004 (2020).","journal-title":"Ann. Math."},{"key":"1430_CR3","doi-asserted-by":"publisher","first-page":"1250024","DOI":"10.1142\/S1230161212500242","volume":"19","author":"T Banica","year":"2012","unstructured":"Banica T., Nechita I., Zyczkowski K.: Almost Hadamard matrices: general theory and examples. Open Syst. Inf. Dyn. 19, 1250024 (2012).","journal-title":"Open Syst. Inf. Dyn."},{"key":"1430_CR4","doi-asserted-by":"publisher","first-page":"627","DOI":"10.4171\/jems\/163","volume":"11","author":"E Bombieri","year":"2009","unstructured":"Bombieri E., Bourgain J.: On Kahane\u2019s ultraflat polynomials. J. Eur. Math. Soc. 11, 627\u2013703 (2009).","journal-title":"J. Eur. Math. Soc."},{"key":"1430_CR5","first-page":"71","volume":"352","author":"P Borwein","year":"2008","unstructured":"Borwein P., Mossinghoff M.: Barker sequences and flat polynomials. London Mathematical Society Lecture Note Series 352, 71\u201388 (2008).","journal-title":"London Mathematical Society Lecture Note Series"},{"key":"1430_CR6","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1112\/jlms\/s2-16.1.76","volume":"2","author":"FW Carroll","year":"1977","unstructured":"Carroll F.W., Eustice D., Figiel T.: The minimum modulus of polynomials with coefficients of modulus one. J. Lond. Math. Soc. 2, 76\u201382 (1977).","journal-title":"J. Lond. Math. Soc."},{"key":"1430_CR7","doi-asserted-by":"publisher","first-page":"2044","DOI":"10.1093\/imrn\/rnad080","volume":"3","author":"X Dong","year":"2024","unstructured":"Dong X., Rudelson M.: Approximately Hadamard matrices and Riesz bases in random frames. Int. Math. Res. Not. 3, 2044\u20132065 (2024).","journal-title":"Int. Math. Res. Not."},{"key":"1430_CR8","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1364\/JOSA.39.000437","volume":"39","author":"MJE Golay","year":"1949","unstructured":"Golay M.J.E.: Multi-slit spectroscopy. J. Opt. Soc. Am. 39, 437\u2013444 (1949).","journal-title":"J. Opt. Soc. Am."},{"key":"1430_CR9","doi-asserted-by":"publisher","DOI":"10.1515\/9781400842902","volume-title":"Hadamard Matrices and Their Applications","author":"KJ Horadam","year":"2007","unstructured":"Horadam K.J.: Hadamard Matrices and Their Applications. Princeton University Press, Princeton (2007)."},{"key":"1430_CR10","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/s10474-015-0517-6","volume":"147","author":"P Jaming","year":"2015","unstructured":"Jaming P., Matolcsi M.: On the existence of flat orthogonal matrices. Acta Math. Hung. 147, 179\u2013188 (2015).","journal-title":"Acta Math. Hung."},{"key":"1430_CR11","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1112\/blms\/12.5.321","volume":"12","author":"J-P Kahane","year":"1980","unstructured":"Kahane J.-P.: Sur les polynomes a coefficients unimodulaires. Bull. Lond. Math. Soc. 12, 321\u2013342 (1980).","journal-title":"Bull. Lond. Math. Soc."},{"key":"1430_CR12","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1016\/S0024-3795(02)00664-X","volume":"367","author":"H Karner","year":"2003","unstructured":"Karner H., Schneid J., Ueberhuber C.: Spectral decomposition of real circulant matrices. Linear Algebra Appl. 367, 301\u2013311 (2003).","journal-title":"Linear Algebra Appl."},{"key":"1430_CR13","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1112\/jlms\/s1-41.1.367","volume":"41","author":"JE Littlewood","year":"1966","unstructured":"Littlewood J.E.: On polynomials $$\\sum ^n \\pm z^m, \\sum ^n e^{\\alpha _m i} z^m, z = e^{\\theta i}$$. J. Lond. Math. Soc. 41, 367\u2013376 (1966).","journal-title":"J. Lond. Math. Soc."},{"key":"1430_CR14","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/s00041-013-9299-9","volume":"20","author":"M Matolcsi","year":"2014","unstructured":"Matolcsi M., Ruzsa I.Z.: Difference sets and positive exponential sums I. General properties. J. Fourier Anal. Appl. 20, 17\u201341 (2014).","journal-title":"J. Fourier Anal. Appl."},{"key":"1430_CR15","unstructured":"Odlyzko A.: Search for ultraflat polynomials with plus and minus one coefficients. In: Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham. Cambridge University Press, Cambridge (2018)."},{"key":"1430_CR16","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1002\/sapm1933121311","volume":"12","author":"R Paley","year":"1933","unstructured":"Paley R.: On orthogonal matrices. J. Math. Phys. 12, 311\u2013320 (1933).","journal-title":"J. Math. Phys."},{"key":"1430_CR17","doi-asserted-by":"crossref","unstructured":"Park K.H., Song H.Y.: Quasi-Hadamard matrix. In: IEEE International Symposium on Information Theory, pp. 1243\u20131247. IEEE (2010).","DOI":"10.1109\/ISIT.2010.5513675"},{"key":"1430_CR18","doi-asserted-by":"publisher","first-page":"855","DOI":"10.1090\/S0002-9939-1959-0116184-5","volume":"10","author":"W Rudin","year":"1959","unstructured":"Rudin W.: Some theorems on Fourier coefficients. Proc. Am. Math. Soc. 10, 855\u2013859 (1959).","journal-title":"Proc. Am. Math. Soc."},{"key":"1430_CR19","doi-asserted-by":"publisher","DOI":"10.5948\/UPO9781614440147","volume-title":"Combinatorial Mathematics","author":"HJ Ryser","year":"1963","unstructured":"Ryser H.J.: Combinatorial Mathematics. Wiley, New York (1963)."},{"key":"1430_CR20","doi-asserted-by":"publisher","first-page":"929","DOI":"10.1090\/S0894-0347-99-00298-2","volume":"12","author":"B Schmidt","year":"1999","unstructured":"Schmidt B.: Cyclotomic integers and finite geometry. J. Am. Math. Soc. 12, 929\u2013952 (1999).","journal-title":"J. Am. Math. Soc."},{"key":"1430_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-59032-5","volume-title":"Orthogonal Designs","author":"J Seberry","year":"2017","unstructured":"Seberry J.: Orthogonal Designs. Springer, Cham (2017)."},{"key":"1430_CR22","doi-asserted-by":"publisher","DOI":"10.1002\/9781119520252","volume-title":"Hadamard Matrices: Constructions Using Number Theory and Algebra","author":"J Seberry","year":"2020","unstructured":"Seberry J., Yamada M.: Hadamard Matrices: Constructions Using Number Theory and Algebra. Wiley, New York (2020)."},{"key":"1430_CR23","unstructured":"Shapiro H.S.: Extremal problems for polynomials. Thesis for S.M. Degree (1952)."},{"key":"1430_CR24","doi-asserted-by":"crossref","unstructured":"Sylvester J.: Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newton\u2019s rule, ornamental tile-work, and the theory of numbers. Lond. Edinb. Dublin Philos. Mag. J. Sci. 34(232), 461\u2013475 (1867).","DOI":"10.1080\/14786446708639914"}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-024-01430-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10623-024-01430-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-024-01430-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,18]],"date-time":"2024-09-18T20:18:52Z","timestamp":1726690732000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10623-024-01430-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,4]]},"references-count":24,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["1430"],"URL":"https:\/\/doi.org\/10.1007\/s10623-024-01430-w","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"value":"0925-1022","type":"print"},{"value":"1573-7586","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,4]]},"assertion":[{"value":"26 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 May 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 May 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 June 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}