Inequalities and bounds for quasi-symmetric 3-designs

AbstractQuasi-symmetric 3-designs with block intersection numbers x and y(0⩽x<y<k) are studied, several inequalities satisfied by the parameters of a quasi-symmetric 3-designs are obtained. Let D be a quasi-symmetric 3-design with the block size k and intersection numbers x, y; y>x⩾1 and suppose D′ denote the complement of D with the block size k′ and intersection numbers x′ and y′. If k −1 ⩽x + y then it is proved that x′ + y′ ⩽ k′. Using this it is shown that the quasi-symmetric 3-designs corresponding to y = x + 1, x + 2 are either extensions of symmetric designs or designs corresponding to the Witt-design (or trivial design, i.e., v = k + 2) or the complement of above designs