In combinatorial mathematics, specifically in combinatorial design theory and combinatorial matrix theory the Williamson conjecture is that Williamson matrices of order exist for all positive integers . Four symmetric and circulant matrices , , , are known as Williamson matrices if their entries are and they satisfy the relationship
where is the identity matrix of order . John Williamson showed that if , , , are Williamson matrices then
is an Hadamard matrix of order . [1] It was once considered likely that Williamson matrices exist for all orders and that the structure of Williamson matrices could provide a route to proving the Hadamard conjecture that Hadamard matrices exist for all orders . [2] However, in 1993 the Williamson conjecture was shown to be false via an exhaustive computer search by Dragomir Ž. Ðoković, who showed that Williamson matrices do not exist in order . [3] In 2008, the counterexamples 47, 53, and 59 were additionally discovered. [4]
In combinatorial mathematics, specifically in combinatorial design theory and combinatorial matrix theory the Williamson conjecture is that Williamson matrices of order exist for all positive integers . Four symmetric and circulant matrices , , , are known as Williamson matrices if their entries are and they satisfy the relationship
where is the identity matrix of order . John Williamson showed that if , , , are Williamson matrices then
is an Hadamard matrix of order . [1] It was once considered likely that Williamson matrices exist for all orders and that the structure of Williamson matrices could provide a route to proving the Hadamard conjecture that Hadamard matrices exist for all orders . [2] However, in 1993 the Williamson conjecture was shown to be false via an exhaustive computer search by Dragomir Ž. Ðoković, who showed that Williamson matrices do not exist in order . [3] In 2008, the counterexamples 47, 53, and 59 were additionally discovered. [4]