Density Evolution, Thresholds and the Stability Condition for Non-binary LDPC Codes

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.