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

The parameters 2- (36, 15, 6) are the smallest parameters of symmetric designs for which a complete classification up to isomorphism is yet unknown. Bouyukliev, Fack and Winne classified all 2- (36, 15, 6) 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- (36, 15, 6) 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.