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.

References

There are many groups of possible parameters for which the existence problem has not been settled.

— Block Designs and K-Geodetic Graphs: A Survey  (2501.09256 - Frasser, 16 Jan 2025) in Introduction, immediately after equation (1)

The following is the list of the remaining 18 sets of parameters for each of which the existence of full pairwise additive BIB designs was still unknown.

— A Method of Construction of Pairwise Additive Balanced Incomplete Block Designs  (2608.21048 - Matsubara et al., 21 Aug 2026) in Section 1, Introduction, Table 1