Some results on the Ryser design conjecture-II

A Ryser design \(\mathcal{D}\) on \(v\) points is a collection of \(v\) proper subsets (called blocks) of a point-set with \(v\) points satisfying (i) every two blocks intersect each other in \(\lambda\) points for a fixed \(\lambda < v\) (ii) there are at least two block sizes. A design \(\mathcal{D}\) is called a symmetric design, if all the blocks of \(\mathcal{D}\) have the same size (or equivalently, every point has the same replication number) and every two blocks intersect each other in \(\lambda\) points. The only known construction of a Ryser design is via block complementation of a symmetric design also known as the Ryser-Woodall complementation method. Such a Ryser design is called a Ryser design of Type-1. The Ryser-Woodall conjecture states: "every Ryser design is of Type-1". Main results of the present article are the following. An expression for the inverse of the incidence matrix \(\mathsf{A}\) of a Ryser design is obtained. A necessary condition for the design to be of Type-1 is obtained. A well known conjecture states that, for a Ryser design on \textit{v} points \(\mbox{ }4\lambda-1\leq v\leq\lambda^2+\lambda+1\). A partial support for this conjecture is obtained. Finally a special case of Ryser designs with two block sizes is shown to be of Type-1.