Frequency-eecient Coding with Low-density Generator Matrices

In a recent paper, it is shown that if Pearl's belief propagation algorithm is applied to the Bayesian belief network of a turbo code, the turbo decoding algorithm immediately results. From this perspective, it is recognized that the turbo coding structure imposes unnecessary diierentiation on the parity checks and the turbo decoding algorithm speciies a seemingly arbitrary sequential activation order for the network. In this paper, a class of high-rate systematic block codes based on pseudo-random low-density generator matrices are proposed as a generalization of the turbo codes in which the parity bits are treated uniformly. These codes can be encoded and decoded in linear time with a parallel distribu-tive decoding algorithm derived from the belief propagation algorithm. We also apply these codes to M-ary modulations using multilevel coding techniques to achieve higher spectral eeciency. In all cases, we have constructed systems with exible error protection capability and performance within 1 dB of the channel capacity.