The Analysis of Error-Correcting Capabilities of Non-Binary LDPC Codes

Non-binary low-density parity-check (NB LDPC) codes significantly outperform their binary counterparts. Moreover, NB LDPC codes are especially good for the channels with burst errors and high-order modulations. This talk is devoted to the analysis of error-correcting capabilities of NB LDPC codes.<br/> We start with distance properties of NB LDPC codes. We consider two approaches to obtain upper bounds for such codes: shortening method and syndrome counting method. Both methods use a sparse structure of the parity-check matrix of NB LDPC codes. We compare obtained upper bounds to the Gallager’s lower bound and to upper bounds for the best non-binary codes. The new upper bounds are shown to lie under the Gilbert-Varshamov bound for some parameters.<br/> The usual way to decode NB LDPC codes is to apply Belief Propagation (BP) decoding. Unfortunately, BP decoding complexity is still large, that is why iterative hard and soft-reliability based decoding majority algorithms are of considerable interest for high-throughput practical applications. In the second part of the talk we investigate the error-correcting capabilities of NB LDPC codes decoded with a hard-decision low-complexity majority algorithm, which is a generalization of the bit-flipping algorithm for binary LDPC codes. We perform the worst-case analysis and estimate the decoding radius realized by this algorithm. By the decoding radius, we mean the number of errors that is guaranteed to be corrected. Our contribution is as follows. We suggest the majority-decoding algorithm with multiple thresholds and give a lower estimate on the decoding radius realized by the new algorithm. The estimate is shown to be at least 1.2 times better than the estimate for a single threshold majority decoder.<br/> At last, we consider BP decoding of NB LDPC codes. We consider an AWGN channel with high-order modulations. In order to reduce the decoding complexity, we propose to use multilevel coding schemes based on NB LDPC codes (NB-LDPC-MLC schemes) over smaller fields. The use of such schemes gives us a reasonable gain in complexity. At the same time the performance of NB-LDPC-MLC schemes is practically the same as the performance of LDPC codes over the field matching the modulation order. In particular, by means of simulations we showed that the performance of NB-LDPC-MLC schemes over GF(16) is the same as the performance of LDPC codes over GF(64) and GF(256) in AWGN channel with QAM 64 and QAM 256 accordingly.