  We discuss questions arising from the recent work of Schellekens, and also
from an earlier paper by Schellekens and Yankielowicz. We summarise
Schellekens' results, and proceed to discuss the uniqueness of the c=24
self-dual conformal field theory with no weight one states, i.e. the Monster
module $V^\natural$. After introducing the concept of complementary
representations, we examine $Z_2$-orbifold constructions in general, and then
proceed to apply our considerations firstly to the specific case of the FKS
constructions $H(\Lambda)$ and then to the reflection twisted theories
$\widetilde H(\Lambda)$. Our techniques provide evidence for the existence of
several new theories beyond those proven to exist in previous work and
conjectured to exist in Schellekens and Yankielowicz.
