Williamson matrices

A Hadamard matrix of order is an -matrix with as entries and such that , where is the transposed matrix of and is the unit matrix of order . Note that the problem of constructing Hadamard matrices of all orders is as yet unsolved (1998; the first open case is ). For a number of methods for constructing Hadamard matrices of concrete orders, see [a1], [a9], [a7]. One of these methods, described below, is due to J. Williamson [a10]. Let , , , and be pairwise commuting symmetric circulant -matrices of order such that (such matrices are called Williamson matrices). Then the Williamson array

is a Hadamard matrix of order . The recent achievements about the construction of Hadamard matrices are connected with the construction of orthogonal designs [a4] (cf. also Design with mutually orthogonal resolutions), Baumert–Hall arrays [a2], Goethals–Seidel arrays [a5] and Plotkin arrays [a6], and with the construction of Williamson-type matrices, i.e., of four or eight -matrices , , of order that satisfy the following conditions:

i) , ;

ii) . Williamson-four matrices have been constructed for all orders , with the exception of , which was eliminated by D.Z. Djokovic [a3], by means of an exhaustive computer search. It is worth mentioning that Williamson-type-four matrices of order are not yet known (1998). Williamson-four and Williamson-type-four matrices are known for many values of . For details, see [a9], Table A1; pp. 543–547. The most recent results can be found in [a11].

There are known Williamson-type-eight matrices of the orders , where , are prime numbers [a8].

A set of -matrices is called a Williamson family, of type , if the following conditions are fulfilled:

a) There exists a -matrix of order such that for arbitrary , ;

b) . If , then the type is denoted by .

If , , and , then each Williamson family of type coincides with a family of Williamson-type matrices.

If , for , and , then each Williamson family of type coincides with a family of Williamson-type-eight matrices.

If , , and , ,

and , then each Williamson family of type coincides with a family of generalized Williamson-type matrices.

An orthogonal design of order and type () on commuting variables is an -matrix with entries from such that

Let be a Williamson family of type and suppose there exists an orthogonal design of type and order that consists of elements , . Then there exists a Hadamard matrix of order . In other words, the existence of orthogonal designs and Williamson families implies the existence of Hadamard matrices. For more details and further constructions see [a4], [a9].

References

[a1]	S.S. Agaian, "Hadamard matrices and their applications" , Lecture Notes Math. , 1168 , Springer (1985)
[a2]	L.D. Baumert, M. Hall Jr., "A new construction for Hadamard matrices" Bull. Amer. Math. Soc. , 71 (1965) pp. 169–170
[a3]	D.Z. Djokovic, "Williamson matrices of order for " Discrete Math. , 115 (1993) pp. 267–271
[a4]	A.V. Geramita, J. Seberry, "Orthogonal designs: Quadratic forms and Hadamard matrices" , M. Dekker (1979)
[a5]	J.M. Goethals, J.J. Seidel, "A skew–Hadamard matrix of order 36" J. Austral. Math. Soc. A , 11 (1970) pp. 343–344
[a6]	M. Plotkin, "Decomposition of Hadamard matrices" J. Combin. Th. A , 2 (1972) pp. 127–130
[a7]	W.D. Wallis, A.P. Street, J.S. Wallis, "Combinatorics: Room squares, sum-free sets and Hadamard matrices" , Lecture Notes Math. , 292 , Springer (1972)
[a8]	J.S. Wallis, "Construction of Williamson type matrices" Linear and Multilinear Algebra , 3 (1975) pp. 197–207
[a9]	J. Seberry, M. Yamada, "Hadamard matrices, sequences and block designs" J.H. Dinitz (ed.) D.R. Stinson (ed.) , Contemporary Design Theory: A Collection of Surveys , Wiley (1992) pp. 431–560
[a10]	J. Williamson, "Hadamard's determinant theorem and the sum of four squares" Duke Math. J. , 11 (1944) pp. 65–81
[a11]	M.Y. Xia, "An infinite class of supplementary difference sets and Williamson matrices" J. Combin. Th. A , 58 (1991) pp. 310–317