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.
Nestings of BIBDs with block size four / Buratti, M.; Kreher, D. L.; Stinson, D. R.. - In: JOURNAL OF COMBINATORIAL DESIGNS. - ISSN 1063-8539. - 32:12(2024), pp. 715-743. [10.1002/jcd.21957]
Nestings of BIBDs with block size four
Buratti M.Primo
Writing – Original Draft Preparation
;
2024
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.| File | Dimensione | Formato | |
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