  We construct Weierstrass data for higher genus embedded doubly periodic
minimal surfaces and present numerical evidence that the associated period
problem can be solved. In the orthogonal ends case, there previously was only
one known surface for each genus. We illustrate multiple new examples for each
genus g>2. In the parallel ends case, the known examples limit as a foliation
of parallel planes with nodes. We construct a new example for each genus g>2
that limit as g-1 singly periodic Scherk surfaces glued between two doubly
periodic Scherk surfaces and also as a singly periodic surface with four
vertical and 2g horizontal Scherk ends.
