Defining sets for 2-(19,9,4) designs and a class of Hadamard designs
Abstract
Seberry [25], Sarvate and Seberry [24] and Kunkle and Seberry [20] have produced defining sets for Hadamard 2-designs on up to 67 elements; these defining sets comprise half of one less than the number of blocks in each design. Smaller defining sets for several of the designs are produced here in two ways: by using the restrictions of the designs on any element and by using the residual designs with respect to any element. The concept of a specifying set for a t-design is introduced to facilitate the production of these defining sets. A known algorithm is adapted slightly to produce all smallest defining sets for the six non-isomorphic 2-(19,9,4) designs. These smallest defining sets are then used to produce all smallest defining sets for the three non-isomorphic 3-(20,10,4) designs. Each smallest defining set for each of these nine designs comprises eight blocks.
