  Large numbers of ground states of two-dimensional Ising spin glasses with
periodic boundary conditions in both directions are calculated for sizes up to
40^2. A combination of a genetic algorithm and Cluster-Exact Approximation is
used. For each quenched realization of the bonds up to 40 independent ground
states are obtained.
  For the infinite system a ground-state energy of e=-1.4015(3) is
extrapolated. The ground-state landscape is investigated using a finite-size
scaling analysis of the distribution of overlaps. The mean-field picture
assuming a complex landscape describes the situation better than the
droplet-scaling model, where for the infinite system mainly two ground states
exist. Strong evidence is found that the ground states are not organized in an
ultrametric fashion in contrast to previous results for three-dimensional spin
glasses.
