Resolvable group divisible designs with block size four and general index

In this paper, we investigate the existence of resolvable group divisible designs (RGDDs) with block size four, group-type h^n and general index @l. The necessary conditions for the existence of such a design are n>=4, hn=0(mod4) and @lh(n-1)=0(mod3). These necessary conditions are shown to be sufficient for all @l>=2, with the definite exceptions of (@l,h,n)@?{(3,2,6)}@?{(2j+1,2,4):j>=1}. The known existence result for @l=1 is also improved.

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

[2]  Gennian Ge,et al.  Super‐simple resolvable balanced incomplete block designs with block size 4 and index 2 , 2007 .

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

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

[7]  Christopher A. Rodger,et al.  Linear spaces with many small lines , 1994, Discret. Math..

[8]  Donald L. Kreher,et al.  A note on {4}-GDDs of type 210 , 2003, Discret. Math..

[9]  Alan C. H. Ling,et al.  A survey on resolvable group divisible designs with block size four , 2004, Discret. Math..

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

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

[12]  Gennian Ge,et al.  Resolvable group divisible designs with block size four and index three , 2006, Discret. Math..

[13]  Gennian Ge,et al.  Asymptotic results on the existence of 4‐RGDDs and uniform 5‐GDDs , 2005 .

[14]  Richard M. Wilson,et al.  On resolvable designs , 2006, Discret. Math..

[15]  Gennian Ge,et al.  Super‐simple resolvable balanced incomplete block designs with block size 4 and index 3 , 2004 .

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

[17]  Christopher A. Rodger,et al.  Existence of certain skew room frames with application to weakly 3-chromatic linear spaces , 1994 .

[18]  Hao Shen,et al.  Resolvable Maximum Packings with Quadruples , 2005, Des. Codes Cryptogr..

[19]  Jiaying Shen,et al.  Existence of resolvable group divisible designs with block size four I , 2002, Discret. Math..

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

[21]  Gennian Ge,et al.  On the existence of partitionable skew Room frames , 2007, Discret. Math..

[22]  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.

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

[24]  Clement W. H. Lam,et al.  Resolvable group divisible designs with block size four and group size six , 2003, Discret. Math..

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

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