Key for these OD's

This is a supplement to the paper Three variable full orthogonal designs of order 56, which will appear in xxx.

All OD(56;s,t,u) where s, t, and u = 56 - s - t are positive are presented. 209 of these were obtained via amicable sets. The 209 can be obtained from 81 OD's of order 56 and of types in up to 8 variables. This frame presents those 81 OD's: OD(56;s,t,u,...)

By definition each of these matrices is an order 56 matrix, OD, with entries from the set {a,-b,b,-b,c,-c,d,-d,e,-e,f,-f,g,-g,h,-h,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 red respectively. Similarly for the other variables. The color for a negative value is a darker variant of the same color used for the positive value. The values for the variables are:
Colour key: a A b B c C d D e E f F g G h H
• 0 is displayed as white in the images (actually 0 never appears in these OD's as the matrices are "full").

These orthogonal designs are constructed from 8 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.

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.

S. Georgiou, W.H. Holzmann, H. Kharaghani & B. Tayfeh-Rezaie, August 2004