Capacity-Achieving Codes for Noisy Channels with Bounded Graphical Complexity and Maximum Likelihood Decoding

In this paper, capacity-achieving codes for memoryless binary-input output-symmetric (MBIOS) channels under maximum-likelihood (ML) decoding with bounded graphical complexity are investigated. The graphical complexity of a code is defined as the number of edges in the graphical representation of the code per information bit and is proportional to the decoding complexity per information bit per iteration under iterative decoding. Irregular repeat-accumulate (IRA) codes are studied first. By deriving their asymptotic average weight distribution (AAWD) it is shown that simple nonsystematic IRA ensembles outperform systematic IRA and regular low-density parity-check (LDPC) ensembles with the same graphical complexity, and are only 0.124 dB away from the Shannon limit for the binary-input additive white Gaussian noise (BIAWGN) channel. However, a conclusive result as to whether these nonsystematic IRA codes can really achieve capacity cannot be reached. Motivated by this inconclusive result, a new family of codes is proposed, called low-density parity-check and generator matrix (LDPC-GM) codes, which are serially concatenated codes with an outer LDPC code and an inner low-density generator matrix (LDGM) code. It is proved that these codes can achieve capacity on any MBIOS channel using ML decoding and also achieve capacity on any BEC using belief propagation (BP) decoding, both with bounded graphical complexity. Moreover, these codes are shown to have linearly increasing minimum distances and achieve the asymptotic Gilbert-Varshamov bound for all rates. 2