Low-density parity-check codes with rates very close to the capacity of the q-ary symmetric channel for large q

Transmission of packets over computer networks is subject to packet-level errors, which appear as "bursts" of bit-level errors and are not well modeled by memoryless binary channels. A standard scrambling technique is used for transmission of packets by the q-ary symmetric channel (q-SC) with alphabet size q and error probability p. Furthermore, significant improvements in computational efficiency can be obtained by codes that operate at the packet-level instead of the bit-level. The capacity of the q-SC is 1+p log_q (p)+(1-p)log_q (1-p)-plog_q (1-q), which is close to 1-p for large q. First designed an efficient decoding algorithm for LDPC codes on the q-SC, and showed that it can afford rates arbitrarily close to 1-2p. We improve the analysis of this decoding algorithm to show that LDPC codes with the Tornado edge distribution, together with an erasure precode, can achieve rates e-close to capacity. We also extend this decoder into a family of decoding algorithms, which become progressively more powerful but also more complex. We show that in the limit, when the decoder is allowed to look infinitely deep into the decoding tree, it can achieve capacity without precoding. However computational requirements make this decoder impractical.