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
*s***a**^{2}+*t***b**^{2}+*u***c**^{2}+... 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
A_{1}, A_{2}, ..., A_{8} where A_{1} is matched with A_{2}, 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*