IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

The search for $\mathrm{BOBIBD}(9,5,5,4,3)$

  • April 15, 2016
  • 2:30 p.m.
  • LeConte 312

Abstract

Given $v$ elements, a \emph{balanced incomplete block design} $\mathrm{BIBD}(v,k,\lambda)$ is a collection of sets of $k$ elements each, called \emph{blocks}, such that each pair of elements occurs in $\lambda$ blocks. A \emph{beautifully ordered balanced incomplete block design} $\mathrm{BOBIBD}(v,k,\lambda,k _ 1,\lambda _ 1)$ is a rectangular array of $v$ elements with the property that the rows form a $\mathrm{BIBD}(v,k,\lambda)$, and the rows of each sub-array formed by taking $k _ 1$ of the columns form a $\mathrm{BIBD}(v,k _ 1,\lambda _ 1)$. Some small examples are easily constructed by hand, and large families can be built from other combinatorial objects such as Latin squares. However, the case of $\mathrm{BOBIBD}(9,5,5,4,3)$ is just small enough not to fall under one of the general methods of construction, and just large enough to be impractical to construct by hand. We do not currently know whether any of these exist. I will describe some of the progress we have made, specifically, cases that can be ruled out, the symmetries of $\mathrm{BIBD}(9,4,3)$'s, and some partial results from computer-based searches.

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