Key for these OD's

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 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 (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 A1, A2, ..., A8 where A1 is matched with A2, 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 A1, A2, A3, A4 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 A1, A2, A3, A4 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.
For 45 OD's including 1,9 there is both an amicable and a negacyclic set. Select either by clicking on the preference above. The two kinds are easily distinguishable from their images: 8 versus 4 subblocks across rows and across columns. Try 1,9 or also, for example, 7,7 versus 7,9.

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