LDPC Decoding Mappings That Maximize Mutual Information

For low-density parity-check (LDPC) codes widely used in NAND flash memories, the bit-error rate performance is closely tied to the number of bits per message used by the message-passing decoder. This paper describes a technique to generate message-passing decoding mapping functions for LDPC codes using 3 and 4 bits per message. These maps are not derived from belief-propagation decoding or one of its approximations, instead, the maps are based on a channel quantizer that maximizes mutual information. More precisely, the construction technique is a systematic method, which uses an optimal quantizer at each step of density evolution to generate message-passing decoding mappings. Numerical results show, for high-rate codes suitable for flash memories, that 4 bits per message and a few iterations (10-20 iterations) are sufficient to approach full belief-propagation decoding, less than 5-7 bits per message typically needed. The construction technique is flexible, since it can generate maps for arbitrary number of bits per message, and can be applied to arbitrary memoryless channels.