Resolvable group divisible designs with block size four and group size six

In this paper, we continue the investigation for the existence of resolvable group divisible designs with block size four, group-type h^n and index unity. The necessary conditions for such a design are n>=4, hn=0(mod4) and h(n-1)=0(mod3). The existence of these designs depends mainly on the cases h=1, 2, 3, 6 and 12. Up to now, more than half of these cases have been solved or almost solved except for h=2 and 6. We shall show that the above necessary conditions are also sufficient for h=6 except n=4 and possibly excepting n@?{6,52,54,58,62,68,74,102,114,124}.

[1]  Gennian Ge.  Uniform frames with block size four and index one or three , 2001 .

[2]  Alan C. H. Ling,et al.  Frames with block size four and index three , 2002 .

[3]  Richard M. Wilson,et al.  Constructions and Uses of Pairwise Balanced Designs , 1975 .

[4]  Hanfried Lenz,et al.  Design theory , 1985 .

[5]  Rolf S. Rees,et al.  Group‐divisible designs with block size k having k + 1 groups, for k = 4, 5 , 2000 .

[6]  W. H. Mills,et al.  Resolvable minimum coverings with quadruples , 1998 .

[7]  Steven Furino,et al.  Frames and Resolvable Designs: Uses, Constructions and Existence , 1996 .

[8]  Rolf S. Rees,et al.  Two new direct product-type constructions for resolvable group-divisible designs , 1993 .

[9]  Alan Hartman,et al.  Resolvable group divisible designs with block size 3 , 1989, Discret. Math..

[10]  Douglas R. Stinson,et al.  Frames with Block Size Four , 1992, Canadian Journal of Mathematics.

[11]  Hao Shen.  On the Existence of Nearly Kirkman Systems , 1992 .

[12]  Richard M. Wilson,et al.  On resolvable designs , 1972, Discret. Math..

[13]  C. Colbourn,et al.  The CRC handbook of combinatorial designs , edited by Charles J. Colbourn and Jeffrey H. Dinitz. Pp. 784. $89.95. 1996. ISBN 0-8493-8948-8 (CRC). , 1997, The Mathematical Gazette.

[14]  Gennian Ge,et al.  Some New uniform frames with block size four and index one or three , 2004 .

[15]  Gennian Ge.  Resolvable group divisible designs with block size four , 2002, Discret. Math..