Jar decoding: LDPC coding theorems for binary input memoryless channels

Recently, a new decoding rule called jar decoding was proposed, under which the decoder first forms a set of suitable size, called a jar, consisting of sequences from the channel input alphabet considered to be closely related to yⁿ, and then takes any codeword from the jar as the estimate of the transmitted codeword. In this paper, we show that under jar decoding, the analysis of low density parity check (LDPC) codes is much easier compared to maximum a posteriori (MAP) or maximum likelihood (ML) and Belief Propagation (BP) decoding, and new general LDPC coding theorems can be established. Specifically, it is proved that LDPC codes can approach the mutual information, with diminishing bit error probability, of any binary input memoryless channel with uniform input distribution when the average variable node degree is large. Moreover, simulation shows an interesting connection between jar decoding and BP decoding, i.e., BP decoding can be regarded as one of many ways to pick up a codeword from the jar for LDPC codes when it succeeds in outputting a codeword.