\(\lambda\)-Designs on \(8p + 1\) Points

Abstract

An \(\lambda\)-design on \(v\) points is a set of \(v\) distinct subsets (blocks) of a \(v\)-element set (points) such that any two different blocks meet in exactly \(\lambda\) points and not all of the blocks have the same size. Ryser’s and Woodall’s \(\lambda\)-design conjecture states that all \(\lambda\)-designs can be obtained from symmetric designs by a certain complementation procedure. The main result of the present paper is that the \(\lambda\)-design conjecture is true when \(v = 8p + 1\), where \(p \equiv 1\) or \(7\) (mod \(8\)) is a prime number.