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Group divisible designs with block size 4 and group sizes 4 and 7

R. Julian R. Abel, Thomas Britz, Yudhistira A. Bunjamin, Diana Combe

Abstract

In this paper, we consider the existence of group divisible designs (GDDs) with block size $4$ and group sizes $4$ and $7$. We show that there exists a 4-GDD of type $4^t 7^s$ for all but a finite specified set of feasible values for $(t, s)$.

Group divisible designs with block size 4 and group sizes 4 and 7

Abstract

In this paper, we consider the existence of group divisible designs (GDDs) with block size and group sizes and . We show that there exists a 4-GDD of type for all but a finite specified set of feasible values for .
Paper Structure (9 sections, 25 theorems, 10 equations, 14 tables)

This paper contains 9 sections, 25 theorems, 10 equations, 14 tables.

Table of Contents

  1. Introduction
  2. Some known results about $4$-GDDs
  3. Designs on sets with $v \le 50$
  4. Existence results for infinite families of types
  5. Some known enumeration results
  6. Some direct constructions
  7. Constructions of $4$-GDDs of type $4^t 7^s$
  8. Summary
  9. Acknowledgements

Key Result

Theorem 1.1

ABC.30lessABC.50lesskrestinreesk=45 Suppose there exists a $4$-GDD of type $\{g_1, g_2, \ldots, g_m\}$ where $g_1 \geq g_2 \geq \cdots \geq g_m >0$. Set $v= \sum_{i=1}^m g_i$. Then

Theorems & Definitions (44)

  • Theorem 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 34 more