In this paper, a modified Belief Propagation (BP) decoding algorithm for low-density parity check (LDPC) codes based on minimum mean square error (MMSE) criterion is proposed. This modified algorithm uses linear equation to replace the hyperbolic function in the original BP algorithm and optimizes the linear approximation error based on MMSE criterion. As a result, compared with the standard BP algorithm the computational complexity is reduced significantly as the modified algorithm requires only addition operations to implement. Besides that simulation results show our modified algorithm can achieve an error performance very close to the BP algorithm on the additive white Gaussian noise channel.

Low-density parity-check (LDPC) codes are one of the most powerful error correcting codes and are attracting much attention these days. LDPC codes are promising for communications and broadcasting as well where the use of error correcting codes are essential. LDPC codes have been standardized in some communication standards, such as, IEEE802.16e, DVB-S2, IEEE802.3an (10BASE-T), and so on. The performance of LDPC codes largely depend on their code structure and decoding algorithm. In this paper, we present the basics of LDPC codes and their decoding algorithms. We also present some LDPC codes that have good performance and are receiving much attention particularly in communication systems. We also overview some standardized LDPC codes, the LDPC codes standardized in DVB-S2 and the IEEE802.16e standard LDPC codes. Moreover, we present some research on LDPC coded MIMO systems and HARQ using LDPC codes.