Inequivalent Hadamard Matrices of Order 40 using Block Designs

This article presents the classification of Hadamard matrices of order 40 with automorphisms of order 19. We determine the 2-(19,9,8) and 2-(20,10,9) designs with automorphisms of order 19. These designs are embedded into 2-(39,19,9) designs which are then extended to 3-(40,20,9) designs. We get 33 non-isomorphic 2-(39,19,9) designs out of which 18 designs gives non-isomorphic 3-(40,20,9) designs and hence 18 inequivalent Hadamard matrices of order 40 with automorphisms of order 19.