Block designs signed over groups of order 2n3m

We introduce a new piecewise construction technique for generalised Bhaskar Rao designs and the concepts of generalised Bhaskar Rao block design pieces and holey generalised Bhaskar Rao block designs. We prove composition theorems for these designs. Using this construction technique and the theory of group representations, and the representations of 2-groups over the field with 3 elements, we show that the established necessary conditions for the existence of generalised Bhaskar Rao designs of block size 3 are sufficient for all groups of order 2n3m.