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By W. D. Wallis, A. P. Street, J. S. Wallis

ISBN-10: 0387060359

ISBN-13: 9780387060354

ISBN-10: 3540060359

ISBN-13: 9783540060352

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Extra info for Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices

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So the adder is always skew. If p n is an odd prime power not of the form 2k+1, then there is a skew Room square o~ s~de pn. The technique of the above theorem cannot be applied to the case of a prime power of the form 2k*l, since i + x 6 = 0 in that case. 4 STRONGSTARTERS FROM STEII~JER TRIPLE SYSTEMS. A S t e i n e r t r i p l e system ~ of order v is simply a balanced incomplete block design on v treatments with parameters k : 3 and ~ : 1. So ~ consists of v(v-l)/6 3-sets (blocks) of the set V of v treatments with the property that for any treatments x and y there is precisely one 3-set containing both x and y.

Starters. Y and Z are easily checked to be Since X is strong, all the xi+Y i are distinct and non-zero. that all the -xi-Y i are distinct and non-zero, so Y is a strong starter. l then xi*Y i = O, so X ~ Z (in fact, X and Z can have no pair in common), and similarly Y # Z. If {xi,Y i} = {-xj,-yj} for some i # j, then xi-Y i = ±(xj-yj), so X would not be a starter; and if {xi,Y i} = {-xi,-y i} then either x i = -Yi - 48 - (impossible as we noticed above), or x. = -x. (impossible in a group of odd order); 1 1 X and Y are not equal.

Therefore, ~ i s a Room square. If we take the intersection of the first n and last v 3 rows with the corresponding columns and delete everything else s we have which is isomorphic t o ~ 2. ~ 3 columns. ,n(vl-l)+l and the corresponding columns. The array formed has entry {0,1i} w h e r ~ {li,l j} w h e r e ~ I had {i,j}, so it is isomorphic t o ~ I. 1 had {O,i} and - 59 - We now assume t h a t ~ 1 a n d ~ 2 were skew and t h a t ~ 3 was a skew subsquare of ~2. The standardization o f ~ can be carried out simply by permuting rows and columns to place ~S in the bottom right corner.

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Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices by W. D. Wallis, A. P. Street, J. S. Wallis

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