Quasi-symmetric designs with good blocks and intersection number one
We show that a quasi-symmetric design with intersection numbers 1 and y > 1 and a good block belongs to one of three types: (a) it has the same parameters as PG(2)(4 q), the design of points and planes in projective 4-space; (b) it is the 2-(23, 7, 21) Witt design; (c) its parameters may be written v = 1 + ((alpha - 1)lambda + 1 (y - 1) and k = 1 + alpha(y - 1), where alpha is an integer and alpha > y greater than or equal to 5, and the design induced on a good block is a 2-(k, y, 1) design. No design of type (c) is known; moreover, for large ranges of the parameters, it cannot exist for arithmetic reasons concerning the parameters. We show also that PG(2)(4, q) is the only design of type (a) in which all blocks are good.
quasi-symmetric designs good blocks intersection numbers