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Designs, Codes and Cryptography

, Volume 57, Issue 1, pp 45–69 | Cite as

On uniformly resolvable designs with block sizes 3 and 4

  • Ernst Schuster
  • Gennian Ge
Article

Abstract

A Uniformly Resolvable Design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k, such a class is denoted k -pc and for a given k the number of k -pcs is denoted r k . In this paper we consider the case of block sizes 3 and 4. The cases r 3 = 1 and r 4 = 1 correspond to Resolvable Group Divisible Designs (RGDD). We prove that if a 4-RGDD of type h u exists then all admissible {3, 4}-URDs with 12hu points exist. In particular, this gives existence for URD with v ≡ 0 (mod 48) points. We also investigate the case of URDs with a fixed number of k -pc. In particular, we show that URDs with r 3 = 4 exist, and that those with r 3 = 7, 10 exist, with 11 and 12 possible exceptions respectively, this covers all cases with 1 < r 3 ≤ 10. Furthermore, we prove that URDs with r 4 = 7 exist and that those with r 4 = 9 exist, except when v = 12, 24 and possibly when v = 276. In addition, we prove that there exist 4-RGDDs of types 2 142, 2 346 and 6 54. Finally, we provide four {3,5}-URDs with 105 points.

Keywords

Uniformly resolvable design Labeled uniformly resolvable design Resolvable group divisible design Frame Transversal design 

Mathematics Subject Classification (2000)

05B05 05B07 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute for Medical Informatics, Statistics and EpidemiologyUniversity of LeipzigLeipzigGermany
  2. 2.Department of MathematicsZhejiang UniversityHangzhouPeople’s Republic of China

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