We present an iterative list decoding algorithm for low-density parity-check (LDPC) codes. In particular we apply this decoder to a class of LDPC codes from finite geometries and show that the (73,45,10) projective geometry code can be maximum-likelihood (ML) decoded with low complexity. Moreover, the list decoding approach enables us to give a theoretical analysis of the performance. We also consider list bit-flipping (BF) decoding of longer LDPC codes.

We analyze the relation between iterative decoding and the properties of the extended parity check matrix. By considering a modified version of bit flipping, which produces a list of decoded words, we derive several relations between decodable error patterns and parameters of the code. By developing a tree of codewords at minimal distance form the received vector, we also obtain new information about the code.