Symmetric 2- ( 36 , 15 , 6 ) $(36,15,6)$ designs with an automorphism of order two

Abstract

Bouyukliev, Fack and Winne classified all 2-$\left(36,15,6\right)$ designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 2-$\left(36,15,6\right)$ designs that admit an automorphism of order two. It is shown that there are exactly $1,547,701$ nonisomorphic such designs, $135,779$ of which are self-dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near-extremal self-dual codes with previously unknown weight distributions.