On the construction of some capacity-approaching coding schemes

Abstract

This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint source-channel coding schemes. We first show some examples of sources and channels where no coding is optimal for all values of the signal~to-noise ratio (SNR). When the source bandwidth is greater than the channel bandwidth, joint coding schemes based on space-filling curves and other families of curves are proposed. For uniform sources and modulo channels, our coding scheme based on space-filling curves operates within 1.1 dB of Shannon's rate-distortion bound. For Gaussian sources and additive white Gaussian noise (AWGN) channels, we can achieve within 0.9 dB of the rate-distortion bound. The second scheme is based on low-density parity-check (LDPC) codes. We first demonstrate that we can translate threshold values of an LDPC code between channels accurately using a simple mapping. We develop some models for density evolution from this observation, namely erasure-channel, Gaussian-capacity, and reciprocal channel approximations. The reciprocal-channel approximation, based on dualizing LDPC codes, provides a very accurate model of density evolution for the AWGN channel. We also develop another approximation method, Gaussian approximation, which enables us to visualize infinite-dimensional density evolution and optimization of LDPC codes. We also develop other tools to better understand density evolution. Using these tools, we design some LDPC codes that approach the Shannon limit extremely closely. For multilevel AWGN channels, we design a rate 1/2 code that has a threshold within 0.0063 dB of the Shannon limit of the noisiest level. For binary input AWGN channels, our best rate 1/2 LDPC code has a .threshold within 0.0045 dB of the Shannon limit. Simulation results show that we can achieve within 0.04 dB of the Shannon limit at a bit error rate of 10-6 using a block length of 107 .