Designs, Codes and Cryptography

, Volume 35, Issue 3, pp 287–302

• 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

References

1. Assaf, A. M., Hartman, A. 1989Resolvable group divisible designs with block size 3 Discrete Math.77520
2. Beth, T, Jungnickel, D, Lenz, H 1985Design TheoryBibliographisches InstitutZurich
3. C. J. Colbourn and J. H. Dinitz, CRC Handbook of Combinatorial Designs, CRC Press Inc., Boca Raton (1996). (New results are reported at http://www.emba.uvm.edu/~dinitz/newresults.html).Google Scholar
4. Colbourn, C. J., Stinson, D. R., Zhu, L. 1997More frames with block size fourJ. Combin. Math. Combin. Comput.23320
5. Furino, S. C., Kageyama, S., Ling, A. C. H., Miao, Y., Yin, J. X. 2002Frames with block size four and index threeJ. Stat. Plann. Infer.106117124
6. Furino, SC, Miao, Y, Yin, JX 1996Frames and Resolvable Designs: Uses, Constructions and ExistenceCRC PressBoca Raton FL
7. Ge, G. 2001Uniform frames with block size four and index one or threeJ. Combin. Designs92839
8. Ge, G. 2002Resolvable group divisible designs with block size fourDiscrete Math.243109119
9. Ge, G., Lam, C. W. H., Ling, A. C. H. 2004Some new uniform frames with block size four and index one or threeJ. Combin. Designs12112122
10. Ge, G., Lam, C. W. H. 2003Resolvable group divisible designs with block size four and group size sixDiscrete Math.268139151
11. Hanani, H., Ray-Chauduri, D. K., Wilson, R. M. 1972On resolvable designsDiscrete Math.3343357
12. Kreher, D. L., Ling, A. C. H., Rees, R. S., Lam, C. W. H. 2003A note on {4}-GDDs of type 210 Discrete Math.261373376
13. Lamken, ER, Mills, WH, Rees, RS 1998Resolvable minimum coverings with quadruplesJ. Combin. Designs6431450
14. Rees, R. S. 1993Two new direct product type constructions for resolvable group divisible designsJ. Combin. Designs11520
15. Rees, R. S. 2000Group divisible designs with block size k having k+1 groups for k= 4 and 5J. Combin. Designs8363386
16. Rees, R. S., Stinson, D. R. 1987On resolvable group-divisible designs with block size 3 Ars Combin.23107120
17. Rees, R. S., Stinson, D. R. 1992Frames with block size fourCanad. J. Math.4410301049
18. Shen, H. 1987Resolvable group divisible designs with block size 4 J. Combin. Math. Combin. Comput.1125130
19. Shen, H. 1992On the existence of nearly Kirkman systemsAnn. Discrete Math.52511518
20. Shen, H. 1996Existence of resolvable group divisible designs with block size four and group size two or threeJ. Shanghai Jiaotong Univ. (Engl. Ed.)16870
21. Shen, H., Shen, J. 2002Existence of resolvable group divisible designs with block size four IDiscrete Math.254513525
22. Stinson, DR 1996PackingsColbourn, CJDinitz, JH eds. CRC Handbook of Combinatorial DesignsCRC PressBoca Raton FLGoogle Scholar
23. Wilson, R. M. 1974Constructions and uses of pairwise balanced designsMath. Centre Tracts551841Google Scholar

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