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

Abstract

Abstract-We propose a decoding algorithm for LDPC codes that achieves the maximum likelihood (ML) solution over the binary erasure channel (BEC). In this channel, the tree-structured expectation propagation (TEP) decoder improves the peeling decoder (PD) by processing check nodes of degree one and two. However, it does not achieve the ML solution, as the tree structure of the TEP allows only for approximate inference. In this paper, we provide the procedure to construct the structure needed for exact inference. This algorithm, denoted as generalized treestructured expectation propagation (GTEP), modifies the code graph by recursively eliminating any check node and merging this information in the remaining graph. The GTEP decoder upon completion either provides the unique ML solution or a tree graph in which the number of parent nodes indicates the multiplicity of the ML solution. We also explain the algorithm as a Gaussian elimination method, relating the GTEP to other ML solutions. Compared to previous approaches, it presents an equivalent complexity, it exhibits a simpler graphical messagepassing procedure and, most interesting, the algorithm can be generalized to other channels.

Message Passing Decoder with Decoding on Zigzag Cycles for Non-binary LDPC Codes

“…Hence the proposed decoding algorithm does not achieve the MAP decoding performance. The generalized tree-structured expectation propagation (GTEP) [12] is an efficient maximum likelihood decoding algorithm † † for the binary LDPC codes. The GTEP carries out the Gaussian elimination if there are no check nodes of degree one in a residual graph.…”

“…The time complexity of the GTEP is roughly approximated by O(|D| 3 ) [12]. The space complexity of the † Notice that j in Step 18 is the unique element of N v (i) \ { j} † † Since we assume that codewords are chosen uniformly, the maximum likelihood decoding achieves the MAP decoding performance.…”

Tree-structured expectation propagation for LDPC decoding over the AWGN channel

“…The asymptotic behavior and scaling laws for the TEP decoder have been also studied in [9]. The generalized TEP (GTEP) decoder, proposed in [10], extends the idea of the TEP decoder by considering relations between any number of variables. Since the GTEP decoder provides the MAP solution over the BEC, it is jointly presented with regular LDPC codes as a capacity achieving scheme [11].…”