This is a supplement to the paper *Three variable full orthogonal designs of order 56*,
Journal of Statistical Planning and Inference, 137 (2007) 611-618.

All OD(56;*s*,*t*,*u*) where *s*, *t*, and *u = 56 - s - t * are positive
are presented.
By definition this matrix is an order 56 matrix, OD, with entries from the set
{**a**,**-b**,**b**,**-b**,**c**,**-c**,**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 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 (actually**0**never appears in these OD's as the matrices are "full").

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
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. For example: 7,7. - They are for a set of 4 negacyclic [right negative shift] matrices A
_{1}, A_{2}, A_{3}, A_{4}which are used in a Goethals-Seidel array, G, of the paper. For example: 7,9. - They are for a set of 4 cyclic matrices A
_{1}, A_{2}, A_{3}, A_{4}which are used in a Goethals-Seidel array, G, of the paper to give an OD of order 28. This OD is then doubled using the doubling lemma. For example: 1,12.

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.

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