On the multiple threshold decoding of LDPC codes over GF(q)

Alexey Frolov; Victor Zyablov

doi:10.3934/amc.2017007

Article Contents

2017, Volume 11, Issue 1: 123-137. Doi: 10.3934/amc.2017007

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On the multiple threshold decoding of LDPC codes over GF(q)

Alexey Frolov^1,2, and
Victor Zyablov^2,

1.
Skolkovo Institute of Science and Technology (Skoltech), and Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia
2.
Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

A part of the results of this paper were presented at the 2015 IEEE International Symposium on Information Theory, June 2015, Hong Kong [7].

Received: July 2015

Revised: March 2016

Published: May 2017

The research was carried out at the IITP RAS and supported by the Russian Science Foundation (project no. 14-50-00150)

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Abstract

We consider decoding of LDPC codes over GF(q) with a harddecision low-complexity majority algorithm, which is a generalization of the bit-flipping algorithm for binary LDPC codes. A modification of this algorithm with multiple thresholds is suggested. A lower estimate on the decoding radius realized by the new algorithm is derived. The estimate is shown to be better than the estimate for a single threshold majority decoder. At the same time, introducing multiple thresholds does not affect the order of decoding complexity.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Tanner graph

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Single threshold decoding. The lower estimate $L(W)$ is valid only up to $W^*$ , that is why the line is dashed after the point. Points A, C and B have coordinates ( $W^*/2$ ; $W^* \ell/2$ ), ( $W^{(S)}$ ; $W^{(S)} \ell$ ) and ( $W^*$ ; $W^* \ell / 2$ ) accordingly

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Figure 3. A subgraph of Tanner graph

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Multiple thresholds. The lower estimate $L(W)$ is valid only up to $W^*$ , that is why the line is dashed after the point

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The dependency of $\alpha^{(S)}$ and $\alpha^{(M)}$ on $\ell$

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Results for $q=16$

( $\ell$ , $n_0$ ); $R$	$\delta$	$\omega^*$	$\rho^{(S)}$	$\rho^{(M)}$	$\rho^{(M)}/\rho^{(S)}$
(45, 52); 0.135	0.6130	0.0103	0.0053	0.0065	1.226
(43, 58); 0.26	0.4855	0.0095	0.0049	0.0060	1.224
(40, 64); 0.375	0.3797	0.0085	0.0044	0.0054	1.227
(31, 62); 0.5	0.2808	0.0072	0.0037	0.0046	1.243
(24, 64); 0.625	0.1935	0.0053	0.0028	0.0034	1.214
(24, 96); 0.75	0.1168	0.0033	0.0017	0.0021	1.235
(26,208);0.875	0.0507	0.0015	0.0008	0.0010	1.250

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Results for $q=64$

( $\ell$ , $n_0$ ); $R$	$\delta$	$\omega^*$	$\rho^{(S)}$	$\rho^{(M)}$	$\rho^{(M)}/\rho^{(S)}$
(21, 24); 0.125	0.7355	0.0156	0.0082	0.0099	1.207
(24, 32); 0.25	0.5863	0.0131	0.0068	0.0083	1.221
(20, 32); 0.375	0.4585	0.0104	0.0054	0.0066	1.222
(22, 44); 0.5	0.3445	0.0081	0.0042	0.0052	1.238
(27, 72); 0.625	0.2415	0.0059	0.0031	0.0038	1.226
(24, 96); 0.75	0.1485	0.0037	0.0019	0.0024	1.263
(26,208); 0.875	0.0661	0.0017	0.0009	0.0011	1.222

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