  A new class of lattice Dirac operators $D$ have been recently proposed on the
basis of the generalized Ginsparg-Wilson relation, $\gamma_5(\gamma_5 D) +
(\gamma_5 D)\gamma_5 =2a^{2k+1}(\gamma_5 D)^{2k+2}$, where $k$ is a
non-negative integer. We discuss the index theorem and locality properties for
this general class of lattice Dirac operators.
