Symmetric 2-(35,17,8) designs with an automorphism of order 2
Sanja Rukavina, Vladimir D. Tonchev
TL;DR
This work completes the classification of symmetric $2$-$(35,17,8)$ designs admitting an automorphism of order $2$, complementing the established results for odd prime orders. Using a four-step, computer-assisted workflow based on the orbit-matrix framework with constraints $\sum_r \gamma_{ir}=k$ and $\sum_r \frac{\Omega_j}{\omega_r}\gamma_{ir}\gamma_{jr}=\lambda \Omega_j+\delta_{ij}(k-\lambda)$, the authors enumerate all feasible action patterns, expand to incidence matrices, and perform isomorphism pruning to obtain 11,642,495 nonisomorphic designs and their automorphism-group structures. They also derive the associated Hadamard $3$-designs, counting 1,015,225 pairwise nonisomorphic $3$-$(36,18,8)$ designs that arise as a derived extension, with complete automorphism-group data. The results substantially advance the understanding of automorphism structures in small symmetric designs and supply extensive enumerative and group-theoretic data, supported by GAP and design-specific software.
Abstract
The largest prime p that can be the order of an automorphism of a 2-(35,17,8) design is p=17, and all 2-(35,17,8) designs with an automorphism of order 17 were classified by Tonchev. The symmetric 2-(35,17,8) designs with automorphisms of odd prime order $p<17$ were also classified. In this paper we give the classification of all symmetric 2-(35,17,8) designs that admit an automorphism of order $p=2$. It is shown that there are exactly $11,642,495$ nonisomorphic such designs. Furthermore, it is shown that the number of nonisomorphic 3-(36,18,8) designs which have at least one derived 2-$(35,17,8)$ design with an automorphism of order 2, is $1,015,225$.