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Designs, Codes and Cryptography

, Volume 35, Issue 3, pp 287–302 | Cite as

Resolvable Maximum Packings with Quadruples

  • Gennian GE
  • C. W. H. LAM
  • Alan C. H. Ling
  • Hao Shen
Article

Abstract

Let V be a finite set of v elements. A packing of the pairs of V by k-subsets is a family F of k-subsets of V, called blocks, such that each pair in V occurs in at most one member of F. For fixed v and k, the packing problem is to determine the number of blocks in any maximum packing. A maximum packing is resolvable if we can partition the blocks into classes (called parallel classes) such that every element is contained in precisely one block of each class. A resolvable maximum packing of the pairs of V by k-subsets is denoted by RP(v,k). It is well known that an RP(v,4) is equivalent to a resolvable group divisible design (RGDD) with block 4 and group size h, where h=1,2 or 3. The existence of 4-RGDDs with group-type h n for h=1 or 3 has been solved except for (h,n)=(3,4) (for which no such design exists) and possibly for (h,n)∈{(3,88),(3,124)}. In this paper, we first complete the case for h=3 by direct constructions. Then, we start the investigation for the existence of 4-RGDDs of type 2 n . We shall show that the necessary conditions for the existence of a 4-RGDD of type 2 n , namely, n ≥ 4 and n ≡ 4 (mod 6) are also sufficient with 2 definite exceptions (n=4,10) and 18 possible exceptions with n=346 being the largest. As a consequence, we have proved that there exists an RP(v,4) for v≡ 0 (mod 4) with 3 exceptions (v=8,12 or 20) and 18 possible exceptions.

Keywords

resolvable packings RGDDs 4-frames 

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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Gennian GE
    • 1
  • C. W. H. LAM
    • 2
  • Alan C. H. Ling
    • 3
  • Hao Shen
    • 4
  1. 1.Department of MathematicsZhejiang UniversityHangzhouP. R. China
  2. 2.Department of Computer ScienceConcordia UniversityMontrealCanada
  3. 3.Department of Computer ScienceUniversity of VermontBurlingtonUSA
  4. 4.Department of MathematicsShanghai Jiao Tong UniversityShanghaiP. R. China

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