Symmetric designs of Menon series admitting an action of Frobenius groups

Mario-Osvin Pavčević, Zagreb, Croatia

Keywords: symmetric design, Frobenius group, orbit structure, index set AMS Subject classification: 05B05

Abstract: In this article we have constructed many designs of Menon series, e.g. symmetric designs with parameters (4u^{2}, 2u^{2} − u, u^{2} − u), assuming that a Frobenius group of order 2u^{2} + u or 2u^{2} − 3u + 1 acts on it in a certain way. It has been proved that there always exist orbit structures for these cases. For smaller values of u, the complete classification has been made. For u = 3, 10 mutually nonisomorphic designs have been constructed. If u = 4, there are 10 nonisomorphic designs and if u = 5, there are 3 nonisomorphic designs. In the case of u = 6, 96 mutually nonisomorphic designs are constructed using this method. The full automorphism groups of these designs have been computed in many cases.