Almost Designs and Their Links with Balanced Incomplete Block Designs

TL;DR

This paper explores a special class of 3-adesigns derived from strongly regular graphs and tournaments, revealing they are also 2-designs, and introduces new classes of 2-adesigns with their properties.

Contribution

It identifies conditions under which certain 3-adesigns are also 2-designs and constructs new classes of 2-adesigns, expanding understanding of their structure and parameters.

Findings

3-adesigns from strongly regular graphs are also 2-designs

Constructed several new classes of 2-adesigns

Discussed restrictions on parameters of these designs

Abstract

Almost designs ($t$-adesigns) were proposed and discussed by Ding as a certain generalization of combinatorial designs related to almost difference sets. Unlike $t$-designs, it is not clear whether $t$-adesigns need also be $(t-1)$-designs or $(t-1)$-adesigns. In this paper we discuss a particular class of 3-adesigns, i.e., 3-adesigns coming from certain strongly regular graphs and tournaments, and find that these are also $2$-designs. We construct several classes of these, and discuss some of the restrictions on the parameters of such a class. We also construct several new classes of 2-adesigns, and discuss some of their properties as well.