Key for these OD's

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.
The two kinds are easily distinguishable from their images: 8 versus 4 subblocks across rows and across columns. Try, for example, 3,16,19 versus 3,16,20.

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