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Group divisible designs with block size 4 and group sizes 2 and 5

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

Abstract

In this paper we provide a $4$-GDD of type $2^2 5^5$, thereby solving the existence question for the last remaining feasible type for a $4$-GDD with no more than $30$ points. We then show that $4$-GDDs of type $2^t 5^s$ exist for all but a finite specified set of feasible pairs $(t,s)$.

Group divisible designs with block size 4 and group sizes 2 and 5

Abstract

In this paper we provide a -GDD of type , thereby solving the existence question for the last remaining feasible type for a -GDD with no more than points. We then show that -GDDs of type exist for all but a finite specified set of feasible pairs .

Paper Structure

This paper contains 9 sections, 21 theorems, 14 equations, 16 tables.

Table of Contents

  1. Introduction
  2. Necessary conditions
  3. A 4-GDD of type $2^2 5^5$
  4. $4$-GDDs of types $g^p$ and $g^p n^1$
  5. $4$-GDDs of type $2^t 5^s$
  6. Direct constructions using automorphism groups
  7. Recursive Constructions
  8. Summary
  9. Acknowledgements

Key Result

Theorem 1.1

5423ABC.50lesskrestinreesk=45 Suppose that there exists a $4$-GDD of type $\{g_1, g_2, \ldots, g_m\}$ and $g_1 \geq g_2 \geq \cdots \geq g_m > 0$. Set $v= \sum_{i=1}^m g_i$. Then

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 23 more