### A SKEW HADAMARD MATRIX OF ORDER 36 [chapter]

J.M. GOETHALS, J.J. SEIDEL
1991 Geometry and Combinatorics
Hadamard matrices exist for infinitely many orders Am, m ^ 1, m integer, including all Am < 100, cf. , . They are conjectured to exist for all such orders. Skew Hadamard matrices have been constructed for all orders Am < 100 except for 36, 52, 76, 92, cf. the table in . Recently Szekeres  found skew Hadamard matrices of the order 2(p'+1) s 12 (mod 16), p prime, thus covering the case 76. In addition, Blatt and Szekeres  constructed one of order 52. The present note contains a
more » ... Hadamard matrix of order 36 (and one of order 52), thus leaving 92 as the smallest open case. The unit matrix of any order is denoted by /. The square matrices Q and R of order m are defined by their only nonzero elements 4;,; + i = <7m, I = 1, » = 1, •••, m -1 ; r, jm _, + 1 = 1, i = 1, • • % m We have g m = 1, R 2 = I, RQ = Q T R. Any square matrix A of order m is symmetric if A = A T , skew if A + A T = 0, circulant if AQ -QA. Hence, for circulant A we have m -l A = £ fljfi 1 , RA = A T R. i = 0 Any square matrix H of order Am is skew Hadamard if its elements are 1 and -1 (we write + and -) and