On the line structure of designs

AbstractA line of a design is the intersection of all the blocks on two points. There is an upper bound on the number of points in a line of a design. A line of a design which meets this upper bound is said to be of maximal length. Restrictions are obtained on the parameters of a symmetric 2-design in order that it might possess a set of lines of maximal length which partitions the point set of the design (that is, a ‘1-spread’). It is shown that the existence of an affine resolvable design with four blocks in each affine resolution class is equivalent to the existence of a Hadamard design possessing a 1-spread. Two recursive constructions for affine resolvable designs with four blocks in each affine resolution class are also given