Design and Analysis of Non-binary LDPC and IRA Modulation Codes

It has been shown that, under belief-propagation (BP) decoding, random-coset GF(q) low-density parity-check (LDPC) codes and irregular repeat-accumulate (IRA) codes with q-ary nonuniform signal constellations approach the unrestricted Shannon limit. Extrinsic information transfer (EXIT) charts are employed in the design of random-coset GF(q) LDPC and IRA modulation codes. However, in the EXIT charts of random-cost GF(q) LDPC and IRA modulation codes, there is no closed-form expression for check node decoder (CND) curves. The CND curves for random-coset GF(q) LDPC and IRA modulation codes rely on Monte Carlo simulations, resulting in high design complexity. This study presents new design methods for random-coset GF(q) LDPC and IRA modulation codes based on average zero-word probability. Average zero-word probability serves as a surrogate for LLR messages, just as mutual information acts as a surrogate for LLR messages in EXIT charts. Based on average zero-word probability, closed-form expressions of CND input-output relations are derived for random-coset GF(q) LDPC and IRA modulation codes. Simple convergent criteria for both random-coset LDPC and IRA modulation codes are proposed. Based on the proposed convergent criteria, six codes are designed with nonuniform signal constellations. Simulation results show that the proposed codes have near-capacity performance and are better than any codes employed by equiprobable uniformly-spaced signal constellations.