{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T19:56:29Z","timestamp":1769198189191,"version":"3.49.0"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"10","license":[{"start":{"date-parts":[[2025,6,28]],"date-time":"2025-06-28T00:00:00Z","timestamp":1751068800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,6,28]],"date-time":"2025-06-28T00:00:00Z","timestamp":1751068800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2025,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We investigate unbiased weighing matrices of weight 9 and provide a construction method using mutually suitable Latin squares. For <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$n \\le 16$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mn>16<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, we determine the maximum size among sets of mutually unbiased weighing matrices of order <jats:italic>n<\/jats:italic> and weight 9. Notably, our findings reveal that 13 is the smallest order where such pairs exist, and 16 is the first order for which a maximum class of unbiased weighing matrices is found.<\/jats:p>","DOI":"10.1007\/s10623-025-01676-y","type":"journal-article","created":{"date-parts":[[2025,6,28]],"date-time":"2025-06-28T03:07:29Z","timestamp":1751080049000},"page":"4235-4273","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Unbiased weighing matrices of weight 9"],"prefix":"10.1007","volume":"93","author":[{"given":"Makoto","family":"Araya","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Masaaki","family":"Harada","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hadi","family":"Kharaghani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sho","family":"Suda","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei-Hsuan","family":"Yu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,6,28]]},"reference":[{"key":"1676_CR1","first-page":"1","volume":"5","author":"M Araya","year":"2018","unstructured":"Araya M., Harada M., Suzuki Y.: Ternary maximal self-orthogonal codes of lengths $$21$$, $$22$$ and $$23$$. J. Algebra Comb. Discret. Struct. Appl. 5, 1\u20134 (2018).","journal-title":"J. Algebra Comb. Discret. Struct. Appl."},{"key":"1676_CR2","doi-asserted-by":"publisher","first-page":"909","DOI":"10.1090\/S0894-0347-07-00589-9","volume":"21","author":"C Bachoc","year":"2008","unstructured":"Bachoc C., Vallentin F.: New upper bounds for kissing numbers from semidefinite programming. J. Am. Math. Soc. 21, 909\u2013924 (2008).","journal-title":"J. Am. Math. Soc."},{"key":"1676_CR3","unstructured":"Bamberg J., Hanaki A., Lansdown J.: AssociationSchemes, A GAP package for working with association schemes and homogeneous coherent configurations, version 3.1.0 (GAP package). http:\/\/www.jesselansdown.com\/AssociationSchemes (2024)."},{"key":"1676_CR4","first-page":"33","volume":"34","author":"EI Bannai","year":"2009","unstructured":"Bannai E.I., Bannai E.T.: On antipodal spherical $$t$$-designs of degree $$s$$ with $$t\\ge 2s-3$$. J. Comb. Inf. Syst. Sci. 34, 33\u201350 (2009).","journal-title":"J. Comb. Inf. Syst. Sci."},{"key":"1676_CR5","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/s10623-014-9944-6","volume":"76","author":"D Best","year":"2015","unstructured":"Best D., Kharaghani H., Ramp H.: Mutually unbiased weighing matrices. Des. Codes Cryptogr. 76, 237\u2013256 (2015).","journal-title":"Des. Codes Cryptogr."},{"key":"1676_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. Comput."},{"key":"1676_CR7","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1112\/S0024611597000403","volume":"75","author":"AR Calderbank","year":"1997","unstructured":"Calderbank A.R., Cameron P.J., Kantor W.M., Seidel J.J.: $$\\mathbb{Z} _4$$-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. Lond. Math. Soc. 75, 436\u2013480 (1997).","journal-title":"Proc. Lond. Math. Soc."},{"key":"1676_CR8","first-page":"299","volume":"21","author":"HC Chan","year":"1986","unstructured":"Chan H.C., Rodger C.A., Seberry J.: On inequivalent weighing matrices. Ars Comb. 21, 299\u2013333 (1986).","journal-title":"Ars Comb."},{"key":"1676_CR9","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1109\/TIT.1979.1056047","volume":"25","author":"JH Conway","year":"1979","unstructured":"Conway J.H., Pless V., Sloane N.: Self-dual codes over $$GF(3)$$ and $$GF(4)$$ of length not exceeding $$16$$. IEEE Trans. Inform. Theory 25, 312\u2013322 (1979).","journal-title":"IEEE Trans. Inform. Theory"},{"key":"1676_CR10","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1007\/BF03187604","volume":"6","author":"P Delsarte","year":"1977","unstructured":"Delsarte P., Goethals J.M., Seidel J.J.: Spherical codes and designs. Geom. Dedic. 6, 363\u2013388 (1977).","journal-title":"Geom. Dedic."},{"key":"1676_CR11","unstructured":"Grant M, Boyd S.: CVX: MATLAB software for disciplined convex programming, Version 2.2. http:\/\/cvxr.com\/cvx (2020)."},{"key":"1676_CR12","unstructured":"Harada M., Munemasa A.: Database of small weighing matrices. https:\/\/www.math.is.tohoku.ac.jp\/~munemasa\/research\/matrices\/weighingmatrices.htm."},{"key":"1676_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00373-025-02912-x","volume":"41","author":"M Harada","year":"2025","unstructured":"Harada M., Kharaghani H.: Unbiased Hadamard matrices and ternary self-dual codes. Graphs Comb. 41, 1\u201310 (2025).","journal-title":"Graphs Comb."},{"key":"1676_CR14","doi-asserted-by":"publisher","first-page":"1063","DOI":"10.1016\/j.jcta.2008.11.011","volume":"116","author":"M Harada","year":"2009","unstructured":"Harada M., Munemasa A.: A complete classification of ternary self-dual codes of length $$24$$. J. Comb. Theory Ser. A 116, 1063\u20131072 (2009).","journal-title":"J. Comb. Theory Ser. A"},{"key":"1676_CR15","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1002\/jcd.20295","volume":"20","author":"M Harada","year":"2012","unstructured":"Harada M., Munemasa A.: On the classification of weighing matrices and self-orthogonal codes. J. Comb. Des. 20, 40\u201357 (2012).","journal-title":"J. Comb. Des."},{"key":"1676_CR16","doi-asserted-by":"crossref","unstructured":"Holzmann W.H., Kharaghani H., Orrick W.: On the real unbiased Hadamard matrices. In: Combinatorics and Graphs, Contemporary Mathematics, vol. 531, pp. 243\u2013250. American Mathematical Society, Providence, RI (2010).","DOI":"10.1090\/conm\/531\/10471"},{"key":"1676_CR17","first-page":"160","volume-title":"Handbook of Combinatorial Designs","author":"R Julian","year":"2007","unstructured":"Julian R., Abel R., Colbourn C.J., Dinitz J.H.: Mutually Orthogonal Latin Squares (MOLS). In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 160\u2013193. Chapman & Hall\/CRC Press, Boca Raton, FL (2007).","edition":"2"},{"key":"1676_CR18","doi-asserted-by":"crossref","unstructured":"Kharaghani H.: Unbiased Hadamard matrices and bases. In: NATO Science for Peace and Security Series D: Information and Communication Security, vol. 29, pp. 312\u2013325. IOS Press, Amsterdam (2011).","DOI":"10.3233\/978-1-60750-663-8-312"},{"key":"1676_CR19","doi-asserted-by":"crossref","first-page":"11","DOI":"10.37236\/2212","volume":"22","author":"H Kharaghani","year":"2015","unstructured":"Kharaghani H., Sasani S., Suda S.: Mutually unbiased Bush-type Hadamard matrices and association schemes. Electron. J. Comb. 22, 11 (2015).","journal-title":"Electron. J. Comb."},{"key":"1676_CR20","doi-asserted-by":"crossref","first-page":"23","DOI":"10.37236\/11701","volume":"31","author":"F-Y Liu","year":"2024","unstructured":"Liu F.-Y., Yu W.-H.: Semidefinite programming bounds for spherical three-distance sets. Electron. J. Comb. 31, 23 (2024).","journal-title":"Electron. J. Comb."},{"key":"1676_CR21","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1137\/0131058","volume":"31","author":"CL Mallows","year":"1976","unstructured":"Mallows C.L., Pless V., Sloane N.: Self-dual codes over $$\\text{ GF } (3)$$. SIAM J. Appl. Math. 31, 649\u2013666 (1976).","journal-title":"SIAM J. Appl. Math."},{"key":"1676_CR22","unstructured":"Niskanen S., \u00d6sterg\u00e5rd P.R.J.: Cliquer user\u2019s guide, version 1.0. Helsinki University of Technology Communications Laboratory Technical Report T48 (2003)."},{"key":"1676_CR23","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1007\/s10801-015-0581-6","volume":"42","author":"H Nozaki","year":"2015","unstructured":"Nozaki H., Suda S.: Weighing matrices and spherical codes. J. Algebr. Comb. 42, 283\u2013291 (2015).","journal-title":"J. Algebr. Comb."},{"key":"1676_CR24","first-page":"161","volume":"5","author":"H Ohmori","year":"1989","unstructured":"Ohmori H.: On the classifications of weighing matrices of order $$12$$. J. Comb. Math. Comb. Comput. 5, 161\u2013215 (1989).","journal-title":"J. Comb. Math. Comb. Comput."},{"key":"1676_CR25","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/0012-365X(93)90394-9","volume":"116","author":"H Ohmori","year":"1993","unstructured":"Ohmori H.: Classification of weighing matrices of order $$13$$ and weight $$9$$. Discret. Math. 116, 55\u201378 (1993).","journal-title":"Discret. Math."},{"key":"1676_CR26","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1109\/TIT.1980.1056195","volume":"26","author":"V Pless","year":"1980","unstructured":"Pless V., Sloane N., Ward H.N.: Ternary codes of minimum weight $$6$$ and the classification of length $$20$$. IEEE Trans. Inform. Theory 26, 305\u2013316 (1980).","journal-title":"IEEE Trans. Inform. Theory"},{"key":"1676_CR27","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1007\/s10623-014-0012-z","volume":"78","author":"S Suda","year":"2016","unstructured":"Suda S.: A two-fold cover of strongly regular graphs with spreads and association schemes of class five. Des. Codes Cryptogr. 78, 463\u2013471 (2016).","journal-title":"Des. Codes Cryptogr."}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-025-01676-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10623-025-01676-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-025-01676-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T18:03:29Z","timestamp":1760551409000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10623-025-01676-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,28]]},"references-count":27,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["1676"],"URL":"https:\/\/doi.org\/10.1007\/s10623-025-01676-y","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"value":"0925-1022","type":"print"},{"value":"1573-7586","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,28]]},"assertion":[{"value":"27 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 April 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 June 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 June 2025","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 there are no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}