  We derive the density evolution equations for non-binary low-density
parity-check (LDPC) ensembles when transmission takes place over the binary
erasure channel. We introduce ensembles defined with respect to the general
linear group over the binary field. For these ensembles the density evolution
equations can be written compactly. The density evolution for the general
linear group helps us in understanding the density evolution for codes defined
with respect to finite fields. We compute thresholds for different alphabet
sizes for various LDPC ensembles. Surprisingly, the threshold is not a
monotonic function of the alphabet size. We state the stability condition for
non-binary LDPC ensembles over any binary memoryless symmetric channel. We also
give upper bounds on the MAP thresholds for various non-binary ensembles based
on EXIT curves and the area theorem.
