By Larry J. Cummings (auth.), Louis R. A. Casse, Walter D. Wallis (eds.)
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Extra resources for Combinatorial Mathematics IV: Proceedings of the Fourth Australian Conference Held at the University of Adelaide August 27–29, 1975
Corollary 2. Suppose there exist ~nicable orthogonal designs of types (ul,u2, .... u t) and (vl,v 2 ..... v s) in order n. Hadamard matrices of order m. Further suppose there exist amicable Then there exist amicable orthogonal designs of types (Ul,(m-l)Ul,mu 2 .... ,mv s) in order mn. Now we see from  there exist amicable Hadamard matrices of order 50 I II III IV V VI 2; r p +i pr(prime power) : 3(mod 4); 2(qS+l) 2(qS+l) qS(prime power) ~ p 2 + ~ 4(qS+l) qS(prime power) ~ p2+36 ~ 5(mod 8); d where d is the product of any of the above orders.
Correspondence between graphs in the first column and these patterns with each peak of height z on a graph corresponding Thus the omitted first graph has no differences treatments different. B~C, but A < C < D in an obvious notation. is very clear~ to a linked subset of z treatments. and the omitted last graph has all 4 The sixth one actually drawn The (in the first column) has A # B, 44 V V v v V V W V v ~V V V way V M A V A/k ~/vk V A A Y AVA A W W W A A v M V k/v A k / w W % kA~ 45 REFERENCES  I.
Since P = J-W*W (* the Hadamard product) is the circulant incidence matrix of the projective plane of order q, P : [ T dl where i D, the set of the di, is a (q2+q+l,q+l,l) planar difference set. Now there exist x,y=x+l in D. Let X = T x, Y = T y and Z = T z for some z eD, z ~ x , z#y. Let m : (q2+q)/2. Then if we use (i) aX + b Y + cW, aX + bY - cW, bl-aT+bT (ii) aX + bY + dW, aX - bY + dW, aX+cY-dW, aX-cY-dW aX+bY, aX-bY, cX + dW, dX - cW (iv) aX + bY + cW, aX + bW - cW, dI + b T - b T 3 + d T 4, (v) aX + b Y +cW, aX + dZ - cW, bY-dZ-cW, (iii) (vi) (vii) (viii) (ix) (x) (xi) 2, aT 2 + b T 3 + d T 4 aX-bY-dZ aX-bY+cZ-dW, aX-bY-cZ+dW cW + dZ + aX + bY, eW + dZ - aX - bY, dW - cZ + aX - bY, dW - cZ - aX + bY aX + bW~ aX - bW, aX+bY+dW, -bX+aY-cW, aX+bY+cZ+dW, aX, bX, -cX-dY+dW, -bX+aY+dZ-cW, cX+dW, cX - dW cX + dW, (xiii) aI + bT TM - bT m+l, bT m + bT m+l , cW, (xiv) aW, bW, cX + dW, d X - cW (xv) aX, bW, cX+dW, dX-cW aX + bW, aX + bW, aX - bW, aW+bX+cY, -dX+cY-dZ+aW cX-dW dX+cW, aX + aY - bW, -dX+cY+aW -cX-dY+aZ+dW, bT m + b T m+l, (xviii) -dI+bT- aX+bY-cZ-dW, a I + b T m - b T m+l, (xvii) 2 aX+bY+cZ+dW, (xii) (xvi) bl+dT-bT aY + bW, cW, aW+bX-cY, cX-dW dW cI + aTm - aT m+l dI bW-aX+dY, bW-aX-dY, respectively, in the Goethals-Seidel array, we obtain the designs of the enunciation.
Combinatorial Mathematics IV: Proceedings of the Fourth Australian Conference Held at the University of Adelaide August 27–29, 1975 by Larry J. Cummings (auth.), Louis R. A. Casse, Walter D. Wallis (eds.)