The stability of LDPC codes over GF(q) with higher order modulation schemes

In this paper, we derive stability conditions for nonbinary low-density-parity-check (LDPC) codes with Gray mapped M-ary phase-shift keying (M-PSK) and M-ary quadrate-amplitude-modulation (M-QAM) constellations. The stability of the GF (q) LDPC decoder with higher order modulation schemes is examined on both additive-white-Gaussian-noise (AWGN) and block Rayleigh fading channels and threshold stability conditions under iterative decoding are obtained using density evolution techniques. It is shown that both on block Rayleigh fading channel when the ideal channel state information (CSI) of the fading process is known, and on AWGN channel, the stability condition of the underlying code is bounded by the Bhattacharya noise parameter. We show that the derived stability condition is both sufficient and necessary condition for iterative belief propagation decoding assuming cycle free massage passing between variable and check nodes. The evolution of stability condition is assessed over several finite fields for M-PSK and M-QAM constellations. This study provides an analytical framework to evaluate the performance bounds of both binary and nonbinary LDPC coded systems that extend over higher order modulation schemes.