Protograph-Based LDPC Hadamard Codes

Abstract

In this paper, we propose a new method to design low-density parity-check Hadamard (LDPC-Hadamard) codesa type of ultimate-Shannon-limit approaching channel codes. The technique is based on applying Hadamard constraints to the check nodes in a generalized protograph-based LDPC code, followed by lifting the generalized protograph. We name the codes formed protograph-based LDPC Hadamard (PLDPC-Hadamard) codes. We also propose a modified Protograph Extrinsic Information Transfer (PEXIT) algorithm for analyzing and optimizing PLDPC-Hadamard code designs. The proposed algorithm further allows the analysis of PLDPC-Hadamard codes with degree-1 and/or punctured nodes. We find codes with decoding thresholds ranging from −1.53 dB to −1.42 dB. At a BER of 10 −5 , the gaps of our codes to the ultimate-Shannon-limit range from 0.40 dB (for rate = 0.0494) to 0.16 dB (for rate = 0.003). Moreover, the error performance of our codes is comparable to that of the traditional LDPC-Hadamard codes. Finally, the BER performances of our codes after puncturing are simulated and compared.

Spatially Coupled PLDPC-Hadamard Convolutional Codes

“…We also develop an efficient decoding algorithm, i.e., a pipeline decoding strategy combined with layered scheduling, for the decoding of SC-PLDPCH-CCs, and analyze its latency and complexity. 3) Using the original PEXIT method in [16], we show that the thresholds of different spatially coupled PLDPC-Hadamard terminated codes (SC-PLDPCH-TDCs) are distinguishable and improves as the coupling length increases. To improve the convergence rate of the original PEXIT method, we propose a layered PEXIT algorithm to efficiently evaluate the threshold of SC-PLDPCH-TDCs with a given coupling length.…”

“…The parity-check matrix derived by the progressive-edge-growth (PEG) method [15] according to the degree distributions does not contain any structure and therefore does not facilitate linear encoding and parallel decoding. In [16], a class of structured LDPC-HCs called protograph-based LDPC-HCs (PLDPC-HCs) have been proposed.…”

“…Using the copy-and-permute operations to lift the protomatrix or protograph, the derived matrix or lifted graph can easily have a quasi-cyclic structure which is conducive to the hardware implementations of encoders and decoders [18]. PLDPC-HCs [16] also inherit such advantages from PLDPC codes. By adding an appropriate amount of degree-1 Hadamard variable nodes (D1H-VNs) to the check nodes in the protograph of a PLDPC code, the SPC constraints are converted into Hadamard constraints and the protograph of a PLDPC-HC is obtained.…”

“…By adding an appropriate amount of degree-1 Hadamard variable nodes (D1H-VNs) to the check nodes in the protograph of a PLDPC code, the SPC constraints are converted into Hadamard constraints and the protograph of a PLDPC-HC is obtained. The protograph-based EXIT (PEXIT) chart method for PLDPC codes [19] has further been modified for analyzing PLDPC-HCs [16]. The modified method not only can produce multiple extrinsic mutual information from the symbol-bysymbol maximum-a-posteriori-probability (symbol-MAP) Hadamard decoder, but also is applicable to analyzing protographs with degree-1 and/or punctured VNs.…”

Protograph Designing of P-LDPC Codes via M3 Method

“…For example, Ref. [16] proposes the Hadamard-based P-LDPC code, Ref. [17] builds non-binary LDPC code by the graphical representation of absorbing sets, and Ref.…”