Document Type
Article
Publication Date
12-2024
Department
Department of Mathematical Sciences
Abstract
In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the nested point) to every block of a (Formula presented.) -BIBD in such a way that we end up with a partial (Formula presented.) -BIBD. In the case where the partial (Formula presented.) -BIBD is in fact a (Formula presented.) -BIBD, we have a perfect nesting. We show that a nesting is perfect if and only if (Formula presented.). Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., (Formula presented.) -BIBDs) when (Formula presented.), as well as for some symmetric BIBDs. Here we study nestings of (Formula presented.) -BIBDs, which are not perfect nestings. We prove that there is a nested (Formula presented.) -BIBD if and only if (Formula presented.), (Formula presented.). This is accomplished by a variety of direct and recursive constructions.
Publication Title
Journal of Combinatorial Designs
Recommended Citation
Buratti, M.,
Kreher, D. L.,
&
Stinson, D.
(2024).
Nestings of BIBDs with block size four.
Journal of Combinatorial Designs,
32(12), 715-743.
http://doi.org/10.1002/jcd.21957
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p2/1136
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Version
Publisher's PDF
Publisher's Statement
© 2024 The Author(s). Journal of Combinatorial Designs published by Wiley Periodicals LLC. Publisher’s version of record: https://doi.org/10.1002/jcd.21957