Existence of balanced incomplete block designs with admissible parameters

Determine whether a balanced incomplete block design exists for every group of parameter values satisfying the necessary relations bk = nr and r(k − 1) = 2(n − 1), thereby resolving the unsettled existence cases for balanced incomplete block designs.

Background

A balanced incomplete block design is specified by parameters (b, n, r, k, λ), where the block size, replication number, and pairwise block-intersection conditions impose necessary arithmetic relations on the parameters. In the paper’s notation, the displayed relations are bk = nr and r(k − 1) = λ(n − 1), with the scanned text rendering λ as 2 in several places.

The paper emphasizes that these relations are necessary but not sufficient: satisfying them does not guarantee that a design exists. It therefore identifies unresolved existence cases among parameter sets that pass the necessary conditions.