Partial geometric designs: Constructions and classifications
Date
2021-08
Authors
Tranel, Theodore J
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Song, Sung-Yell
Hartwig, Jonas
Herzog, David
Lutz, Jack
Martin, Ryan
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Abstract
We study partial geometric designs in detail and investigate their connection with ordinary $t$-designs, partial geometries, transversal designs, partial geometric difference sets and families, directed strongly regular graphs, and other tactical configurations. We survey and investigate various construction methods and characterizations of partial geometric designs, and try to give a list of all important existence results.
We show that all partial geometric designs can be divided into a (small) number of concurrence profiles according to the nature of the concurrences among the points. We try to classify the known partial geometric designs according to their concurrence profile and the spectrum of the concurrence matrix. Simple and important concurrence profiles involved with well-known incidence structures, such as $2$-designs, partial geometries, and transversal designs (TDs) are characterized. Namely, (i) an ordinary $2$-$(v,k,\lambda)$ design has a single concurrence $\lambda$, (ii) a partial geometry with parameters $(\kappa, \rho, \tau)$ has two concurrences $1$ and $0$, and (iii) a TD$_\lambda(k,u)$ has two concurrences $\lambda$ and $0$. We then show (iv) the existence of other partial geometric designs having two concurrences, say $\lambda_1$ and $\lambda_2$, $(\lambda_1 > \lambda_2 >0)$, and (v) the existence of partial geometric designs with three concurrences, $\lambda_1 > \lambda_2 > \lambda_3 \geq 0$. We give a list of partial geometric designs in the cases of (iv) and (v). In particular, we find all partial geometric designs of order up to 12 whose concurrence matrices are circulant according to their spectra and concurrence profiles. We then give the methods of constructing these partial geometric designs along with the analysis of combinatorial properties.
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dissertation