Good Codes Based on Very Sparse Matrices

Unfortunately this paper contains two errors with respect to Gallager's work on low density parity check codes. We gained the impression from the literature that \the sparse parity check codes studied by Gallager are bad," but this is in fact not the case. We also had the impression that Gallager's decoding algorithm was the same as Meier and Staaelbach's, and that our use of belief propagation was a new innovation. However, Gallager in fact proposed and used the identical belief propagation algorithm in 1962. We became aware of these errors shortly before the IMA conference on Cryptography and Coding (December 1995). We established that Gallager's low density parity check codes share all thègoodness' properties of thèMN' codes presented in this paper, and that their empirical performance is superior, as described in our more recent papers.