{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:28:15Z","timestamp":1787326095649,"version":"3.56.0"},"reference-count":31,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>We prove the existence of $(-1,1)$-matrices with near-extremal properties. In particular, we find matrices having either small inner products between all pairs of distinct rows or having determinants approaching Hadamard's bound. Our approach is to study the $m \\times n$ submatrices of a Hadamard matrix of order k. We use both the probabilistic method to obtain information about randomly selected submatrices and also combinatorial bounds which hold for all submatrices. Our existence results depend upon the gaps between successive primes. Beyond the applications presented in this paper, the study of randomly selected submatrices of combinatorial designs is of intrinsic interest.<\/jats:p>","DOI":"10.1137\/09075192x","type":"journal-article","created":{"date-parts":[[2009,9,23]],"date-time":"2009-09-23T18:09:54Z","timestamp":1253729394000},"page":"1422-1440","source":"Crossref","is-referenced-by-count":3,"title":["$(1,-1)$-Matrices with Near-Extremal Properties"],"prefix":"10.1137","volume":"23","author":[{"given":"Warwick","family":"de Launey","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David A.","family":"Levin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,9,23]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/83.3.532"},{"key":"R2","first-page":"70","volume":"71","author":"Barba G.","year":"1933","journal-title":"Giorn. Mat. Battaglini"},{"key":"R3","unstructured":"R. Craigen and H. Kharaghani,\n                      Hadamard matrices and Hadamard designs\n                      , in Handbook of Combinatorial Designs, 2nd ed., Discrete Math. Appl. (Boca Raton), C. J. Colbourn and J. H. Dinitz, eds., Chapman & Hall\/CRC, Boca Raton, FL, 2007, pp. 273\u2013279."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.4064\/aa-2-1-23-46"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(99)00229-2"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2009.01.001"},{"key":"R7","unstructured":"W. de Launey and D. M. Gordon,\n                      On the Distribution of Known Hadamard Matrices\n                      , manuscript, 2009."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01109911"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01111249"},{"key":"R10","first-page":"30","volume":"37","author":"Gilman R. E.","year":"1931","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"A. Granville,\n                      Unexpected irregularities in the distribution of prime numbers\n                      , in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z\u00fcrich, 1994), Birkh\u00e4user, Basel, 1995, pp. 388\u2013399.","DOI":"10.1007\/978-3-0348-9078-6_32"},{"key":"R12","first-page":"240","volume":"17","author":"Hadamard J.","year":"1893","journal-title":"Bull. Sci. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0007-4497","issn-type":"print"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01214174"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177731236"},{"key":"R15","first-page":"39","volume":"29","author":"Koukouvinos C.","year":"2000","journal-title":"Bull. Inst. Combin. Appl."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(01)00249-X"},{"key":"R17","unstructured":"F. J. MacWilliams and N. J. A. Sloane,\n                      The Theory of Error-Correcting Codes.\n                      I, North\u2013Holland Mathematical Lib. 16, North\u2013Holland, Amsterdam, 1977."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1990-0972703-X"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"C. McDiarmid,\n                      Concentration\n                      , in Probabilistic Methods for Algorithmic Discrete Mathematics, Algorithms Combin. 16, Springer, Berlin, 1998, pp. 195\u2013248.","DOI":"10.1007\/978-3-662-12788-9_6"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177730883"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/0022-314X(86)90101-0"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(96)00147-4"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2006.04.041"},{"key":"R24","unstructured":"W. P. Orrick, B. Solomon, R. Dowdeswell, and W. D. Smith,\n                      New Lower Bounds for the Maximal Determinant Problem\n                      , http:\/\/arXiv.org\/abs\/math\/0304410 (2003)."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1002\/sapm1933121311"},{"key":"R26","first-page":"87","volume":"47","author":"Selberg A.","year":"1943","journal-title":"Arch. Math. Naturvid."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1907-01572-0"},{"key":"R28","unstructured":"R. P. Stanley,\n                      Enumerative Combinatorics\n                      , Vol. 1, Cambridge Stud. Adv. Math. 49, Cambridge University Press, Cambridge, UK, 1997."},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20109"},{"key":"R30","first-page":"95","volume":"59","author":"Tur\u00e1n P.","year":"1940","journal-title":"Math. Naturwiss. Anz. Ungar. Akad. Wiss."},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.4064\/cm-12-1-73-83"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/09075192X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:26:48Z","timestamp":1787322408000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/09075192X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":31,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/09075192X"],"URL":"https:\/\/doi.org\/10.1137\/09075192x","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}