Tree-structure expectation propagation for decoding LDPC codes over binary erasure channels

Abstract

Abstract-Expectation Propagation is a generalization to Belief Propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pair-wise marginal distribution constraints in some check nodes of the LDPC Tanner graph. These additional constraints allow decoding the received codeword when the BP decoder gets stuck. In this paper, we first present the new decoding algorithm, whose complexity is identical to the BP decoder, and we then prove that it is able to decode codewords with a larger fraction of erasures, as the block size tends to infinity. The proposed algorithm can be also understood as a simplification of the Maxwell decoder, but without its computational complexity. We also illustrate that the new algorithm outperforms the BP decoder for finite block-size codes.

Tree Expectation Propagation for ML Decoding of LDPC Codes over the BEC

“…By including also pairwise factors, i.e. w i,j (V i , V pi ) = w i,j (V i , V pi ), we have the TEP decoder in [31], [32]. In this section, we briefly review these two algorithms for BEC as a starting point for our GTEP decoder.…”

“…The TEP decoder [31], [32] can be understood as an extension of the PD that processes, each iteration, either a degree-one or a degree-two check node. A degree-two check node tells us that the variables connected to it are either equal, if the parity check is zero, or opposite, if the parity check is one.…”

“…This algorithm, hereafter referred to as tree-structured EP or TEP, allows capturing some of the information provided by the interactions between the variables in the graph. In a novel work, we have put forward the TEP decoder for LDPC codes over the BEC [31], [32], borrowing from the tree-structured EP algorithm. The TEP decoder is able to continue decoding when the BP gets stuck, since it solves relations between two variables.…”

“…The TEP decoder is able to continue decoding when the BP gets stuck, since it solves relations between two variables. It is proven in [31], [32] that the TEP exhibits an improvement with respect to the performance of the BP, but not fully achieving the MAP solution. The TEP only considers relations between two variables to ensure that the degree of the check nodes does not increase.…”

Finite-length analysis of the TEP decoder for LDPC ensembles over the BEC

“…In Fig. 3, we compare the estimated TEP expected graph evolution (in solid lines) for R 1 (τ ) = r 1 (τ )E and R 2 (τ ) = r 2 (τ )E for a regular (3,6) code with 20 particular decoding trajectories obtained through simulation (thin lines). We choose a very large code length, n = 2 17 , so that the deviation around the expected evolution is small.…”