Construction of new symmetric designs with parameters (66,26,10)

✍ Scribed by I. Matulić-Bedenić; K. Horvatić-Baldasar; E. Kramer

• Publisher: John Wiley and Sons
• Year: 1995
• Tongue: English
• Weight: 245 KB
• Volume: 3
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.3180030605

✦ Synopsis

It is known that there exists only one (Tran van Trung's) design for (66,26,10) up to now. In this article we consider designs for (66,26,10) with the Frobenius group F39 and we prove that there exist (up to isomorphism) exactly 18 such designs. 0 1995 John Wiley & Sons, h e . In 1982, Tran van Trung [3] constructed a (self-dual) symmetric design with parameters (66,26,10) whose full automorphism group was F55 X 2 2 (the direct product of the Frobenius group F55 of order 55 with a cyclic group 2 2 of order 2). Hence the full automorphism group has order 110 and so is "big" in the sense of Aschbacher [ l ] because the order of the group is not smaller than the number of points which is 66.

Since this time no further such designs have been constructed. In particular, it is not known if there is a symmetric design with parameters (66,26,10) with a "small" automorphism group.

We prove here the following result.

Theorem. We suppose that the Frobenius group F39 of order 39 operates on a symmetric design with parameters (66,26, lo), where 2 1 3 has exactly 1 jixed point and Z, has exactly 6 jixed points. Then there exist (up to isomorphism) exactly 18 such designs.