36×36 {-1, +1} matrices of maximal determinant

Ratio of |Det R_j| to Hadamard bound: 1

M=R_j^TR_j= R_j R_j^T=36 I
where I is the 36×36 identity matrix.

Notes:

1. Maximal matrix first reported by Paley [P].
2. For a list of Ted Spence's 180 Hadamard matrices of order 36 related to regular two-graphs on 36 vertices [Sp4], see his home page. The 24 order 36 Hadamard matrices of Goethals-Seidel type found by Spence and Turyn are listed there as well. The former set is also available at Jennifer Seberry's home page along with 15 additional matrices. A single file containing all these matrices, with duplicates removed, is available from the home page of Christos Koukouvinos. He also provides a separate file containing only skew Hadamard matrices from the above list.
3. Jennifer Seberry additionally gives three Williamson-type matrices and one good matrix of order 36. The first and third Williamson matrix from her list are equivalent to the two matrices available in Sloane's Library of Hadamard Matrices. The first is also equivalent to the matrix obtained from Paley's (second) construction.
4. The 80 inequivalent (15,35,7,3,1)-designs (available from designtheory.org; such designs are examples of Steiner triple systems) give rise to 80 inequivalent Hadamard matrices [GS]. See also [CMW], [WSW] pg. 343, and [Ca]. These matrices form a subset of the 180 matrices on Spence's list of matrices related to regular two-graphs.
5. The 22 inequivalent Latin squares of order 6 (available from Brendan McKay's web site) give rise to 11 inequivalent Hadamard matrices [GS]. See also [CMW] and [Ca]. These matrices form a subset of the 180 matrices on Spence's list of matrices related to regular two-graphs.
6. The list of 235 matrices analyzed in [O2] includes all of the above, as well as some additional matrices collected from the literature and other sources. The lower bound on the number of equivalence classes of Hadamard matrices of order 36 given in [O2] is 3,734,467.
7. Iliya Bouyukliev, Veerle Fack, and Joost Winne have reported the complete classification of (35,17,8)-designs with an automorphism of order 3 fixing 2 points and blocks. These give rise to 7238 inequivalent Hadamard matrices [BFW].