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Enumeration of \(t\)-designs through intersection matrices. (English) Zbl 1046.05012

Summary: We exploit some intersection matrices to empower a backtracking approach based on Kramer-Mesner matrices. As an application, we consider the family of simple \(t\)-\((t + 8,t + 2,4)\) designs, \(1 \leq t \leq 4\), and provide a complete classification for \(t = 1,4\), as well as a classification of all non-rigid designs for \(t = 2,3\). We also enumerate all rigid designs for \(t = 2\). The computations confirm the results obtained in B. C. Denny and R. Mathon [J. Stat. Plann. Inference 106, 5–19 (2002; Zbl 1127.05301)] through the new approach which is much simpler. Finally a list of other designs constructed by this method is provided.

MSC:

05B05 Combinatorial aspects of block designs
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05E20 Group actions on designs, etc. (MSC2000)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures

Citations:

Zbl 1127.05301

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