A note on a decoding algorithm of codes on graphs with small loops

The best-known algorithm for the decoding of low-density parity-check (LDPC) codes is the sum-product algorithm (SPA). The SPA is a message-passing algorithm on a graphical model called a factor graph (FG). The performance of the SPA depends on a structure of loops in a FG. Pearl showed that loops in a graphical model could be erased by the clustering method. This method clusters plural nodes into a single node. In this paper, we show several examples about a decoding on a FG to which the clustering method is applied. And we propose an efficient decoding algorithm for it. For a binary erasure channel (BEC), the performance with this method goes up clearly.