Optimized degree distributions for binary and non-binary LDPC codes in Flash memory

This paper uses extrinsic-information-transfer (EXIT)-function analysis employing the reciprocal channel approximation (RCA) to obtain optimal LDPC code degree distributions for initial hard decoding (one-bit quantization of the channel output) and for decoding with the soft information provided by additional reads in both SLC (two-level cell) and MLC (four-level-cell) Flash memory. These results indicate that design for hard decoding can provide irregular degree distributions that have good thresholds across the range of possible decoding precisions. These results also quantify the potential benefit of irregular LDPC codes as compared to regular LDPC codes in the flash setting and compare the RCA-EXIT thresholds of word-line voltages optimized for maximum mutual information (MMI) and word-line voltages that explicitly minimize the RCA-EXIT threshold of a specific LDPC degree distribution. Along the way, this paper illustrates that the MMI optimization of word-line voltages for five reads is a quasi-convex problem for the Gaussian model of SLC Flash. The paper also uses RCA-based EXIT analysis to show that for the same spectral efficiency of 0.9 bits per cell, rate-0.45 non-binary LDPC codes on MLC Flash systems provide thresholds more than 0.5 dB better than rate-0.9 binary LDPC codes on SLC Flash with the same number of reads (i.e. three reads that would provide hard decisions for MLC and limited soft information for SLC). The MLC approach has a potential threshold improvement of more than 1.5 dB over the SLC approach when both systems have access to the full soft information.