A Ryser design has equally many points as blocks with the provision that every two blocks intersect in a fixed number of points A. An improper Ryser design has only one replication number and is thus symmetric design. A proper Ryser design has two replication numbers. The only known construction of a Ryser design is the complementation of a symmetric design. Such a Ryser design is called a Ryser design of type 1. Let D denote a Ryser design of order v, index A and replication numbers r(1). r(2). Let e(i) denote the number of points of 13 with replication number r(i) (with i = 1, 2). Call a block A small (respectively large) if vertical bar A vertical bar 2 lambda) and average if vertical bar A vertical bar = 2 lambda. Let D denote the integer e(1) - r(2) and let rho > 1 denote the rational number r(1)-1/r(2)-1 Main results of the present article are the following. For every block A, r(1) >= vertical bar A vertical bar >= r(2) (this improves an earlier known inequality vertical bar A vertical bar >= r(2)). If there is no small block (respectively no large block) in D, then D = 0). With an extra assumption e(2) > e(1) an earlier known upper bound on v is improved from a cubic to a quadratic in lambda. It is also proved that if v <= lambda(2) + lambda + 1 and if rho equals lambda or lambda - 1, then D is of type 1. Finally, a Ryser design with 2(n) + 1 points is shown to be of type 1.
展开▼
