Biblio

A new framework is presented in this paper for proving coding theorems for linear codes, where the systematic bits and the corresponding parity-check bits play different roles. Precisely, the noisy systematic bits are used to limit the list size of typical codewords, while the noisy parity-check bits are used to select from the list the maximum likelihood codeword. This new framework for linear codes allows that the systematic bits and the parity-check bits are transmitted in different ways and over different channels. In particular, this new framework unifies the source coding theorems and the channel coding theorems. With this framework, we prove that the Bernoulli generator matrix codes (BGMCs) are capacity-achieving over binary-input output symmetric (BIOS) channels and also entropy-achieving for Bernoulli sources.

We present a ternary source coding scheme in this paper, which is a special class of low density generator matrix (LDGM) codes. We prove that a ternary linear block LDGM code, whose generator matrix is randomly generated with each element independent and identically distributed, is universal for source coding in terms of the symbol-error rate (SER). To circumvent the high-complex maximum likelihood decoding, we introduce a special class of convolutional LDGM codes, called block Markov superposition transmission of repetition (BMST-R) codes, which are iteratively decodable by a sliding window algorithm. Then the presented BMST-R codes are applied to construct a tandem scheme for Gaussian source compression, where a dead-zone quantizer is introduced before the ternary source coding. The main advantages of this scheme are its universality and flexibility. The dead-zone quantizer can choose a proper quantization level according to the distortion requirement, while the LDGM codes can adapt the code rate to approach the entropy of the quantized sequence. Numerical results show that the proposed scheme performs well for ternary sources over a wide range of code rates and that the distortion introduced by quantization dominates provided that the code rate is slightly greater than the discrete entropy.