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Skew-orthogonal steiner triple systems

✍ Scribed by P. Dukes; E. Mendelsohn

Publisher: John Wiley and Sons
Year: 1999
Tongue: English
Weight: 302 KB
Volume: 7
Category: Article
ISSN: 1063-8539
DOI: 10.1002/(sici)1520-6610(1999)7:6<431::aid-jcd4>3.0.co;2-#

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✦ Synopsis

Two Steiner triple systems, S 1 VY B 1 and S 2 VY B 2 , are orthogonal (S 1 c S 2 ) if B 1 B 2 Y and if fuY vg T fxY yg, uvwY xyw P B 1 , uvsY xyt P B 2 then s T t. The solution to the existence problem for orthogonal Steiner triple systems, (OSTS) was a major accomplishment in design theory. Two orthogonal triple systems are skew-orthogonal (SOSTS, written S 1 v S 2 ) if, in addition, we require uvwY xys P B 1 and uvtY xyw P B 2 implies s T t.

Orthogonal triple systems are associated with a class of Room squares, with the skew orthogonal triple systems corresponding to skew Room squares. Also, SOSTS are related to separable weakly union-free TTS. SOSTS are much rarer than OSTS; for example SOSTSv do not exist for v 3Y 9Y 15. Furthermore, a fundamental construction for the earlier OSTS proofs when v 3 mod 6 cannot exist. In the case v 1 mod 6 we are able to show existence except possibly for 22 values, the largest of which is 1315. There are at least two nonisomorphic OSTS19s one of which is SOSTS19 and the other not. A SOSTS27 was found, implying the existence of SOSTSv for v 3 mod 6 with ®nitely many possible exceptions.

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