engleski

On some recent results on 2-(v, k, λ) symmetric designs with small λ

Fundamental problems of design theory are those of existence and classification of designs with certain parameter set. In this talk we are interested in biplanes and triplanes, i.e., in 2- (v, k, 2) and 2-(v, k, 3) symmetric designs. The existence of a biplane with parameters (121, 16, 2) is an open problem. We show that such a biplane do not have an automorphism of order 11 or 13. Further, we exclude a possible action of some small groups of order divisible by five or seven, on a biplane with parameters (121, 16, 2). Triplanes of order 12, i.e. symmetric (71, 15, 3) designs, have the greatest number of points among all known triplanes and it is not known if a triplane (v, k, 3) exists for v > 71. All 146 previously known (71, 15, 3) designs admit an action of an automorphism of order 3. We give the first example of a triplane of order 12 that does not admit an automorphism of order 3, obtained by using binary linear codes.

symmetric design ; biplane ; triplane

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