Fixed block configuration group divisible designs with block size six

TL;DR

This paper investigates the existence and construction of group divisible designs with block size six and two groups, focusing on fixed configurations of points from each group, providing new existence results and explicit constructions.

Contribution

It establishes sufficiency conditions for certain configurations and offers new constructions for various group sizes and configurations.

Findings

Necessary conditions are sufficient for configuration (3,3).

Constructs minimal or near-minimal index examples for (1,5) configuration.

Provides constructions for several families with (2,4) configuration.

Abstract

We present constructions and results about GDDs with two groups and block size 6. We study those GDDs in which each block has configuration (s,t), that is in which each block has exactly s points from one of the two groups and t points from the other. We show the necessary conditions are sufficient for the existence of GDD(n,2,6;{\lambda}1,{\lambda}2)s with fixed block configuration (3,3). For configuration (1,5), we give minimal or near-minimal index examples for all group sizes n \geq 5 except n = 10, 15, 160, or 190. For configuration (2,4), we provide constructions for several families of GDD(n,2,6;{\lambda}1,{\lambda}2)s.