Editing Hadamard matrix (section)

==Hadamard conjecture==
{{unsolved|mathematics|Is there a Hadamard matrix of order 4''k'' for every positive integer ''k''?}}
The most important [[open problem|open question]] in the theory of Hadamard matrices is one of existence. Specifically, the '''Hadamard conjecture''' proposes that a Hadamard matrix of order 4''k'' exists for every positive integer ''k''. The Hadamard conjecture has also been attributed to Paley, although it was considered implicitly by others prior to Paley's work.<ref>{{cite journal
 | last1 = Hedayat | first1 = A.
 | last2 = Wallis | first2 = W. D.
 | issue = 6
 | journal = [[Annals of Statistics]]
 | jstor = 2958712
 | mr = 523759
 | pages = 1184–1238
 | title = Hadamard matrices and their applications
 | volume = 6
 | year = 1978
 | doi=10.1214/aos/1176344370
| doi-access = free
 }}.</ref>

A generalization of Sylvester's construction proves that if <math>H_n</math> and <math>H_m</math> are Hadamard matrices of orders ''n'' and ''m'' respectively, then <math>H_n \otimes H_m</math> is a Hadamard matrix of order ''nm''. This result is used to produce Hadamard matrices of higher order once those of smaller orders are known.

Sylvester's 1867 construction yields Hadamard matrices of order 1, 2, 4, 8, 16, 32, etc. Hadamard matrices of orders 12 and 20 were subsequently constructed by Hadamard (in 1893).<ref>{{cite journal |first=J. |last=Hadamard |title=Résolution d'une question relative aux déterminants |journal=[[Bulletin des Sciences Mathématiques]] |volume=17 |pages=240–246 |year=1893 }}</ref> In 1933, [[Raymond Paley]] discovered the [[Paley construction]], which produces a Hadamard matrix of order ''q'' + 1 when ''q'' is any [[prime power]] that is [[modular arithmetic|congruent]] to 3 modulo 4 and that produces a Hadamard matrix of order 2(''q'' + 1) when ''q'' is a prime power that is congruent to 1 modulo 4.<ref>{{cite journal |first=R. E. A. C. |last=Paley |title=On orthogonal matrices |journal=[[Journal of Mathematics and Physics]] |volume=12 |issue= 1–4|pages=311–320 |year=1933 |doi= 10.1002/sapm1933121311}}</ref> His method uses [[finite field]]s.

The smallest order that cannot be constructed by a combination of Sylvester's and Paley's methods is 92. A Hadamard matrix of this order was found using a computer by [[Leonard Baumert|Baumert]], [[Solomon W. Golomb|Golomb]], and [[Marshall Hall (mathematician)|Hall]] in 1962 at [[JPL]].<ref>{{cite journal |first1=L. |last1=Baumert |first2=S. W. |last2=Golomb |first3=M. Jr. |last3=Hall |title=Discovery of an Hadamard Matrix of Order 92 |journal=[[Bulletin of the American Mathematical Society]] |volume=68 |issue=3 |pages=237–238 |year=1962 |doi=10.1090/S0002-9904-1962-10761-7 |mr=0148686 |doi-access=free }}</ref> They used a construction, due to [[John Williamson (mathematician)|Williamson]],<ref>{{cite journal |first=J. |last=Williamson |title=Hadamard's determinant theorem and the sum of four squares |journal=[[Duke Mathematical Journal]] |volume=11 |issue=1 |pages=65–81 |year=1944 |doi=10.1215/S0012-7094-44-01108-7 |mr=0009590 }}</ref> that has yielded many additional orders. Many other methods for constructing Hadamard matrices are now known.

In 2005, Hadi Kharaghani and Behruz Tayfeh-Rezaie published their construction of a Hadamard matrix of order 428.<ref>{{cite journal |first1=H. |last1=Kharaghani |first2=B. |last2=Tayfeh-Rezaie |title=A Hadamard matrix of order 428 |journal=Journal of Combinatorial Designs |volume=13 |year=2005 |issue=6 |pages=435–440 |doi=10.1002/jcd.20043 |s2cid=17206302 }}</ref> As a result, the smallest order for which no Hadamard matrix is presently known is 668. <!-- Anon contributor: please go to the article's talk page and discuss your objection to this claim; properly sourced material cannot be removed from Wikipedia without a good reason. -->

By 2014, there were 12 multiples of 4 less than 2000 for which no Hadamard matrix of that order was known.<ref name="dokovic">{{Cite journal| doi=10.1002/jcd.21358| last1=Đoković| first1=Dragomir Ž| last2=Golubitsky| first2=Oleg | last3=Kotsireas |first3=Ilias S. |title=Some new orders of Hadamard and Skew-Hadamard matrices| journal=Journal of Combinatorial Designs| year=2014| volume=22| issue=6|pages=270–277|  arxiv=1301.3671| s2cid=26598685}}</ref> They are:
668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, and 1964.