2024/04/03 by Marco Buratti, Buratti, Marco, Donald L. Kreher +3 · 1 citation
Computer Science · #05B05 #Advanced Data Storage Technologies #Algorithms and Data Compression #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2404.02666
openalex publication_date 2024/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (v,k,λ)-BIBD in such a way that we end up with a partial (v,k+1,λ+1)-BIBD. In the case where the partial (v,k+1,λ+1)-BIBD is in fact a (v,k+1,λ+1)-BIBD, we have a perfect nesting. We show that a nesting is perfect if and only if k = 2 λ+ 1. Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., (v,3,1)-BIBDs) when v ≡ 1 \bmod 6, as well as for some symmetric BIBDs. Here we study nestings of (v,4,1)-BIBDs, which are not perfect nestings. We prove that there is a nested (v,4,1)-BIBD if and only if v ≡ 1 or 4 \bmod 12, v ≥ 13. This is accomplished by a variety of direct and recursive constructions.