Good error-correcting codes based on very sparse matrices

We report theoretical and empirical properties of Gallager´s (1963) low density parity check codes on Gaussian channels. It can be proved that, given an optimal decoder, these codes asymptotically approach the Shannon limit. With a practical `belief propagation´ decoder, performance substantially better than that of standard convolutional and concatenated codes can be achieved; indeed the performance is almost as close to the Shannon limit as that of turbo codes