Symmetric 2- ( 35 , 17 , 8 ) Designs With an Automorphism of Order 2

The largest prime $p$ that can be the order of an automorphism of a 2-$\left(35,17,8\right)$ design is $p=17$, and all 2-$\left(35,17,8\right)$ designs with an automorphism of order 17 were classified by Tonchev. The symmetric 2-$\left(35,17,8\right)$ designs with automorphisms of an odd prime order $p<17$ were classified in Bouyukliev, Fack and Winne and Crnković and Rukavina. In this paper we give the classification of all symmetric 2-$\left(35,17,8\right)$ designs that admit an automorphism of order $p=2$. It is shown that there are exactly $11\text{,642,}495$ nonisomorphic such designs. Furthermore, it is shown that the number of nonisomorphic 3-$\left(36,18,8\right)$ designs which have at least one derived 2-$\left(35,17,8\right)$ design with an automorphism of order 2, is $1,015,225$.