  An exact algorithm is used to investigate the distributions of the
degeneracies of low-energy excited states for the bimodal Ising spin glass on
the brickwork lattice. Since the distributions are extreme and do not
self-average, we base our conclusions on the most likely values of the
degeneracies. Our main result is that the degeneracy of the first excited state
per ground state and per spin is finite in the thermodynamic limit. This is
very different from the same model on a square lattice where a divergence
proportional to the linear lattice size is expected. The energy gap for the
brickwork lattice is obviously 2J on finite systems and predicted to be the
same in the thermodynamic limit. Our results suggest that a 2J gap is universal
for planar bimodal Ising spin glasses. The distribution of the second
contribution to the internal energy has a mode close to zero and we predict
that the low-temperature specific heat is dominated by the leading term
proportional to $T^{-2} \exp(-2J/kT)$.
