Low-density parity-check codes

Low-density parity-check (LDPC) codes are among the most efficient error correcting codes in use. This chapter introduces an important family of LDPC ensembles, based on random factor graphs, and studies some of their basic properties. It focuses on performances under optimal decoding, when no constraint is imposed on the computational complexity of the decoding procedure. Bounds in their performances are derived through an analysis of the geometric properties of their codebook. In particular, it shows that appropriately chosen LDPC ensembles allow for communication reliably at rates close to Shannon's capacity.