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 orthogonal designs are constructed from either 8 or 4 matrices. The first rows of these component matrices are displayed as follows:

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