Design and Analysis of Non-Binary LDPC and IRA Modulation Codes Using Average Zero-Word Probability

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. In previous works, extrinsic information transfer (EXIT) charts were employed in the design of random-coset GF(q) LDPC and IRA modulation codes. However, in the EXIT charts, 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 a high design complexity. This study presents new design methods of random-coset GF(q) LDPC and IRA modulation codes based on the average zero-word probability. The average zero-word probability serves as a surrogate for LLR messages, just as the mutual information acts as a surrogate for LLR messages in EXIT charts. Based on the 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 convergence criteria for random-coset LDPC and IRA modulation codes are proposed. Based on the proposed convergence criteria, six codes are designed with nonuniform signal constellations. Simulation results show that the proposed codes have near-capacity performances and are comparable with those designed based on EXIT charts.