This is a supplement to the paper All triples for orthogonal designs of order 40, Discrete Math, 308 (2008), pages 2796 - 2801, which appears in a special volume of Discrete Math honouring Jennifer Seberry on her 60th birthday.

All OD(40;s,t,u) where s, t, u are positive and s+t+u is at most 40 are presented. By definition this matrix is an order 40 matrix, OD, with entries from the set {a,-b,b,-b,c,-c,0}, such that OD times its transpose is sa²+tb²+uc² times the identity matrix.

• The first variable, which occurs s times per row, is represented as a or A = -a. These are displayed in the image as light or dark yellow respectively.
• The second variable, which occurs t times per row, is represented as b or B = -b. These are displayed in the image as light or dark red respectively.
• The third variable, which occurs u times per row, is represented as c or C = -c. These are displayed in the image as light or dark blue respectively.
• 0 is displayed as white in the images.

The orthogonal designs are constructed from either 8 or 4 matrices. The first rows of these component matrices are displayed as follows:

• They are for an amicable set of 8 cyclic matrices given in order A₁, A₂, ..., A₈ where A₁ is matched with A₂, etc. which are used in the Kharaghani array, K, of the paper.
• They are for a set of 4 negacyclic [right negative shift] matrices A₁, A₂, A₃, A₄ which are used in a Goethals-Seidel array, G, of the paper.

In the first case, if the matching is special, then infinite classes of designs are constructible from these. See Theorem 4 in the paper of H. Kharaghani: Arrays for orthogonal designs, J. Combin. Des., 8 (2000), 127-130. Select special matchings to have them displayed.

W.H. Holzmann, H. Kharaghani & B. Tayfeh-Rezaie, June 2003