Induction theorems for generalized Bhaskar Rao designs.
Adrian M. Nelson
Abstract
There are extensive results known for the existence of
generalized Bhaskar Rao designs signed over solvable groups,
and particularly for designs with block size . There
have so far been no comparable results for any non-solvable
groups and in particular none for the non-solvable group of
smallest order, the simple group . In this
paper we define the new notion of pairwise balanced signed
block designs, signed over a group. Our central new result is
then a composition theorem for these pairwise balanced
signed block designs. From this we derive a pair of induction
theorems specifically for constructing generalized Bhaskar Rao
design pieces. These induction theorems give conditions under
which generalized Bhaskar Rao designs pieces signed over a group
can be induced from such designs signed over a subgroup. This
is in contrast to long established results which give
conditions under which generalized Bhaskar Rao design pieces
signed over a quotient group can be inflated to give such
designs signed over the whole group. By making systematic use of
our new induction theorems and various piecewise
constructions we are able to elegantly establish that the well
known necessary condition for the existence of generalized
Bhaskar Rao designs of block size are also sufficient
for designs signed over the non-solvable groups
, and
. In the course of these applications we
identify a number of new generic generalized Bhaskar Rao design
pieces. Finally, and independently of the new induction
theorems, we identify a new infinite family of solvable groups
for which the known necessary conditions for the existence of
generalized Bhaskar Rao designs of block size are also
sufficient.
Keywords:
Generalized Bhaskar Rao designs, difference matrices, group divisible designs, holey generalized Bhaskar Rao designs.
AMS Subject Classification:
Primary 05B05; secondary 05B10, 05B30, 51E05.