# Hybrid LDPC codes and iterative decoding methods - i3s

Hybrid LDPC codes and iterative decoding methods - i3s

List of figures

1.1 Parity-check matrix of a non-binary LDPC code and its bipartite graph. . 26

1.2 Representation of a ensemble of irregular LDPC codes. . . . . . . . . . . 28

1.3 Variable node update . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.4 Check node update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.5 EXIT curves of (2, 4) GF(2), GF(8) and GF(256) regular codes. The

SNR is 0.7dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.1 Factor graph of parity-check of an hybrid LDPC code. . . . . . . . . . . 46

2.2 Message transform through linear map. . . . . . . . . . . . . . . . . . . 48

2.3 Parametrization of a hybrid LDPC code ensemble . . . . . . . . . . . . . 51

2.4 Quantities Ω for hybrid and non-hybrid LDPC codes in terms of maximum

symbol order q max . These figures show that a hybrid LDPC code

can be stable when a non-binary code is not. . . . . . . . . . . . . . . . . 59

2.5 FER versus E b

N o

: code rate one-half. K = 1024 information bits except for

the multi-edge type LDPC code for which K = 1280 information bits.

No finite length optimization has been applied. N iter = 500 except for

quasi-cyclic LDPC code (from [1]) for which N iter = 50. . . . . . . . . . 76

2.6 FER versus E b

N o

(in dB): code rate one-half. N bit = 2048 coded bits except

for the multi-edge type LDPC code for which N bit = 2560 coded bits.

N iter = 500 decoding iterations are performed. . . . . . . . . . . . . . . 77

2.7 Comparison of hybrid LDPC code with Turbo Hadamard codes (TH)

taken from [2] and Zigzag Hadamard (ZH) codes taken from [3], for an

information block length of K bit ≃ 200. N iter = 30 for Turbo Hadamard

codes, and N iter = 200 for the hybrid LDPC codes. . . . . . . . . . . . . 79

2.8 Comparison of hybrid LDPC code with punctured Turbo Hadamard (PTH)

taken from [4] and other powerful codes, for code rate 1/6. The PTH code

has K bit = 999 information bits, and the other codes have K bit = 1024

information bits. N iter = 50 for the PTH code, and N iter = 200 for the

other codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.1 General definition of a formal neuron . . . . . . . . . . . . . . . . . . . 97

3.2 An artificial neuron which computes the weighted sum of the inputs, and

the apply the activation function f. . . . . . . . . . . . . . . . . . . . . . 98

3.3 A polynomial neuron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

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