Gallager introduced a family of codes based on sparse bipartite graphs, which he called low-density parity-check codes. He suggested a natural decoding algorithm for these codes, and proved a good bound on the fraction of errors that can be corrected. As the codes that Gallager builds are derived from regular graphs, we refer to them as regular codes.
We consider error-correcting codes based on random irregular bipartite graphs, which we call irregular codes. We introduce tools based on linear programming for designing linear time irregular codes with better error-correcting capabilities than possible with regular codes. For example, the decoding algorithm for the rate 1/2 regular codes of Gallager can provably correct up to 5.17% errors asymptotically, whereas we have found irregular codes for which our decoding algorithm can provably correct up to 6.27% errors asymptotically. We include the results of simulations demonstrating the effectiveness of our codes on systems of reasonable size.
Originally appeared in the Proceedings of the 30th ACM Symposium on Theory of Computing, pp. 249--258, 1998.