A New Framework for Proving Coding Theorems for Linear Codes

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.