On the multiple threshold decoding of LDPC codes over GF(<i>q</i>)

Previous Article
Generalized Hamming weights of codes over the $\mathcal{GH}$ curve
AMC Home
This Issue
Next Article
AFSRs synthesis with the extended Euclidean rational approximation algorithm

February 2017, 11(1): 123-137. doi: 10.3934/amc.2017007

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 February 2017

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

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: Coding theory, iterative decoding, LDPC codes, majority logic decoding, threshold decoding, decoding radius.

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

Citation: Alexey Frolov, Victor Zyablov. On the multiple threshold decoding of LDPC codes over GF(q). Advances in Mathematics of Communications, 2017, 11 (1) : 123-137. doi: 10.3934/amc.2017007

References:

[1]	N. Alon, Eigenvalues and expanders, Combinarorica, 6 (1986), 83-96. doi: 10.1007/BF02579166. Google Scholar
[2]	A. Barg and A. Mazumdar, On the number of errors correctable with codes on graphs, IEEE Trans. Inf. Theory, 57 (2011), 910-919. doi: 10.1109/TIT.2010.2094812. Google Scholar
[3]	J. Boutros, O. Pothier and G. Zémor, Generalized low density (Tanner) codes, in Proc. IEEE Int. Conf. Comm. , Vancouver, 1999,441-445. doi: 10.1109/ICC.1999.767979. Google Scholar
[4]	D. Burshtein, On the error correction of regular LDPC codes using the flipping algorithm, IEEE Trans. Inf. Theory, 54 (2008), 517-530. doi: 10.1109/TIT.2007.913261. Google Scholar
[5]	M. C. Davey and D. MacKay, Low-density parity check codes over GF(q), IEEE Commun. Lett., 2 (1998), 165-167. doi: 10.1109/ITW.1998.706440. Google Scholar
[6]	A. Frolov and V. Zyablov, Asymptotic estimation of the fraction of errors correctable by q-ary LDPC codes, Probl. Inf. Transm., 46 (2010), 142-159. doi: 10.1134/S0032946010020043. Google Scholar
[7]	A. Frolov and V. Zyablov, On the multiple threshold decoding of LDPC codes over GF(q), in Proc. 2015 IEEE Int. Symp. Inf. Theory, Hong Kong, 2015,2673-2677. doi: 10.1109/ISIT.2015.7282941. Google Scholar
[8]	R. G. Gallager, Low-Density Parity-Check Codes, MIT Press, Cambridge, 1963. doi: 10.1109/TIT.1962.1057683.
[9]	F. Garcia-Ferrero, D. Declercq and J. Valls, Non-binary LDPC decoder based on symbol flipping with multiple votes, IEEE Commun. Lett., 18 (2014), 749-752. Google Scholar
[10]	N. Kahale, On the second eigenvalue and linear expansion of regular graphs, in Proc. IEEE Symp. Found. Comp. Sci. , 1992,296-303. doi: 10.1109/SFCS.1992.267762. Google Scholar
[11]	S. Kovalev, Decoding of low-density codes, Probl. Inf. Transm., 27 (1991), 51-56. Google Scholar
[12]	M. Lentmaier and K. Zigangirov, On generalized low-density parity-check codes based on Hamming component codes, IEEE Commun. Lett., 3 (1999), 248-250. doi: 10.1109/4234.781010. Google Scholar
[13]	A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinarorica., 8 (1988), 261-277. doi: 10.1007/BF02126799. Google Scholar
[14]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, NorthHolland, Amsterdam, 1991.Google Scholar
[15]	G. A. Margulis, Explicit constructions of concentrators, Probl. Inform. Transm., 9 (1975), 325-332. Google Scholar
[16]	J. Massey, Threshold Decoding, MIT Press, Cambridge, 1963.
[17]	P. Rybin, On the error-correcting capabilities of low-complexity decoded irregular LDPC codes, in Proc. 2014 IEEE Int. Symp. Inf. Theory, Honolulu, 2014,3165-3169. doi: 10.1109/ISIT.2014.6875418. Google Scholar
[18]	M. Sipser and D. A. Spielman, Expander codes, IEEE Trans. Inf. Theory, 42 (1996), 1710-1722. doi: 10.1109/18.556667. Google Scholar
[19]	H. Song and J. R. Cruz, Reduced-complexity decoding of Q-ary LDPC codes for magnetic recording, IEEE Transact. Magnetics, 39 (2003), 1081-1087. Google Scholar
[20]	R. Tanner, A recursive approach to low complexity codes, IEEE Trans. Inf. Theory, 27 (1981), 533-547. doi: 10.1109/TIT.1981.1056404. Google Scholar
[21]	J. Webber, T. Nishimura, T. Ohgane and Y. Ogawa, A study on adaptive thresholds for reduced complexity bit-flip decoding, in Proc. Int. Conf. Adv. Commun. Techn. , 2012,497-501.Google Scholar
[22]	J. Webber, T. Nishimura, T. Ohgane and Y. Ogawa, Performance investigation of reduced complexity bit-flipping using variable thresholds and noise perturbation, in Proc. Int. Conf. Adv. Commun. Techn. , 2014,206-2013. doi: 10.1109/ICACT.2014.6779175. Google Scholar
[23]	K. Zigangirov, A. Pusane, D. Zigangirov and D. Costello, On the error-correcting capability of LDPC codes, Probl. Inf. Transm., 44 (2008), 214-225. doi: 10.1134/S0032946008030046. Google Scholar
[24]	V. Zyablov, R. Johannesson and M. Lonċar, Low-complexity error correction of Hammingcode-based LDPC codes, Probl. Inf. Trans., 45 (2009), 95-109. doi: 10.1134/S0032946009020021. Google Scholar
[25]	V. Zyablov and M. Pinsker, Estimation of the error-correction complexity for Gallager lowdensity codes, Probl. Inf. Transm., 11 (1975), 18-28. Google Scholar

show all references

References:

[1]	N. Alon, Eigenvalues and expanders, Combinarorica, 6 (1986), 83-96. doi: 10.1007/BF02579166. Google Scholar
[2]	A. Barg and A. Mazumdar, On the number of errors correctable with codes on graphs, IEEE Trans. Inf. Theory, 57 (2011), 910-919. doi: 10.1109/TIT.2010.2094812. Google Scholar
[3]	J. Boutros, O. Pothier and G. Zémor, Generalized low density (Tanner) codes, in Proc. IEEE Int. Conf. Comm. , Vancouver, 1999,441-445. doi: 10.1109/ICC.1999.767979. Google Scholar
[4]	D. Burshtein, On the error correction of regular LDPC codes using the flipping algorithm, IEEE Trans. Inf. Theory, 54 (2008), 517-530. doi: 10.1109/TIT.2007.913261. Google Scholar
[5]	M. C. Davey and D. MacKay, Low-density parity check codes over GF(q), IEEE Commun. Lett., 2 (1998), 165-167. doi: 10.1109/ITW.1998.706440. Google Scholar
[6]	A. Frolov and V. Zyablov, Asymptotic estimation of the fraction of errors correctable by q-ary LDPC codes, Probl. Inf. Transm., 46 (2010), 142-159. doi: 10.1134/S0032946010020043. Google Scholar
[7]	A. Frolov and V. Zyablov, On the multiple threshold decoding of LDPC codes over GF(q), in Proc. 2015 IEEE Int. Symp. Inf. Theory, Hong Kong, 2015,2673-2677. doi: 10.1109/ISIT.2015.7282941. Google Scholar
[8]	R. G. Gallager, Low-Density Parity-Check Codes, MIT Press, Cambridge, 1963. doi: 10.1109/TIT.1962.1057683.
[9]	F. Garcia-Ferrero, D. Declercq and J. Valls, Non-binary LDPC decoder based on symbol flipping with multiple votes, IEEE Commun. Lett., 18 (2014), 749-752. Google Scholar
[10]	N. Kahale, On the second eigenvalue and linear expansion of regular graphs, in Proc. IEEE Symp. Found. Comp. Sci. , 1992,296-303. doi: 10.1109/SFCS.1992.267762. Google Scholar
[11]	S. Kovalev, Decoding of low-density codes, Probl. Inf. Transm., 27 (1991), 51-56. Google Scholar
[12]	M. Lentmaier and K. Zigangirov, On generalized low-density parity-check codes based on Hamming component codes, IEEE Commun. Lett., 3 (1999), 248-250. doi: 10.1109/4234.781010. Google Scholar
[13]	A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinarorica., 8 (1988), 261-277. doi: 10.1007/BF02126799. Google Scholar
[14]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, NorthHolland, Amsterdam, 1991.Google Scholar
[15]	G. A. Margulis, Explicit constructions of concentrators, Probl. Inform. Transm., 9 (1975), 325-332. Google Scholar
[16]	J. Massey, Threshold Decoding, MIT Press, Cambridge, 1963.
[17]	P. Rybin, On the error-correcting capabilities of low-complexity decoded irregular LDPC codes, in Proc. 2014 IEEE Int. Symp. Inf. Theory, Honolulu, 2014,3165-3169. doi: 10.1109/ISIT.2014.6875418. Google Scholar
[18]	M. Sipser and D. A. Spielman, Expander codes, IEEE Trans. Inf. Theory, 42 (1996), 1710-1722. doi: 10.1109/18.556667. Google Scholar
[19]	H. Song and J. R. Cruz, Reduced-complexity decoding of Q-ary LDPC codes for magnetic recording, IEEE Transact. Magnetics, 39 (2003), 1081-1087. Google Scholar
[20]	R. Tanner, A recursive approach to low complexity codes, IEEE Trans. Inf. Theory, 27 (1981), 533-547. doi: 10.1109/TIT.1981.1056404. Google Scholar
[21]	J. Webber, T. Nishimura, T. Ohgane and Y. Ogawa, A study on adaptive thresholds for reduced complexity bit-flip decoding, in Proc. Int. Conf. Adv. Commun. Techn. , 2012,497-501.Google Scholar
[22]	J. Webber, T. Nishimura, T. Ohgane and Y. Ogawa, Performance investigation of reduced complexity bit-flipping using variable thresholds and noise perturbation, in Proc. Int. Conf. Adv. Commun. Techn. , 2014,206-2013. doi: 10.1109/ICACT.2014.6779175. Google Scholar
[23]	K. Zigangirov, A. Pusane, D. Zigangirov and D. Costello, On the error-correcting capability of LDPC codes, Probl. Inf. Transm., 44 (2008), 214-225. doi: 10.1134/S0032946008030046. Google Scholar
[24]	V. Zyablov, R. Johannesson and M. Lonċar, Low-complexity error correction of Hammingcode-based LDPC codes, Probl. Inf. Trans., 45 (2009), 95-109. doi: 10.1134/S0032946009020021. Google Scholar
[25]	V. Zyablov and M. Pinsker, Estimation of the error-correction complexity for Gallager lowdensity codes, Probl. Inf. Transm., 11 (1975), 18-28. Google Scholar

Figure 1. Tanner graph

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. 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

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. A subgraph of Tanner graph

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. Multiple thresholds. The lower estimate $L(W)$ is valid only up to $W^*$, that is why the line is dashed after the point

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. The dependency of $\alpha^{(S)}$ and $\alpha^{(M)}$ on $\ell$

Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. 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

Table Options

Download as excel

($\ell$, $n_0$); $R$	$\delta$	$\omega^*$	$\rho^{(S)}$	$\rho^{(M)}$	$\rho^{(M)}/\rho^{(S)}$
(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

Table 2. 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

Table Options

Download as excel

($\ell$, $n_0$); $R$	$\delta$	$\omega^*$	$\rho^{(S)}$	$\rho^{(M)}$	$\rho^{(M)}/\rho^{(S)}$
(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

[1]	Kwankyu Lee. Decoding of differential AG codes. Advances in Mathematics of Communications, 2016, 10 (2) : 307-319. doi: 10.3934/amc.2016007
[2]	Hannes Bartz, Antonia Wachter-Zeh. Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases. Advances in Mathematics of Communications, 2018, 12 (4) : 773-804. doi: 10.3934/amc.2018046
[3]	Johan Rosenkilde. Power decoding Reed-Solomon codes up to the Johnson radius. Advances in Mathematics of Communications, 2018, 12 (1) : 81-106. doi: 10.3934/amc.2018005
[4]	Elisa Gorla, Felice Manganiello, Joachim Rosenthal. An algebraic approach for decoding spread codes. Advances in Mathematics of Communications, 2012, 6 (4) : 443-466. doi: 10.3934/amc.2012.6.443
[5]	Washiela Fish, Jennifer D. Key, Eric Mwambene. Partial permutation decoding for simplex codes. Advances in Mathematics of Communications, 2012, 6 (4) : 505-516. doi: 10.3934/amc.2012.6.505
[6]	Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433
[7]	Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385
[8]	Heide Gluesing-Luerssen, Uwe Helmke, José Ignacio Iglesias Curto. Algebraic decoding for doubly cyclic convolutional codes. Advances in Mathematics of Communications, 2010, 4 (1) : 83-99. doi: 10.3934/amc.2010.4.83
[9]	Joan-Josep Climent, Diego Napp, Raquel Pinto, Rita Simões. Decoding of $2$D convolutional codes over an erasure channel. Advances in Mathematics of Communications, 2016, 10 (1) : 179-193. doi: 10.3934/amc.2016.10.179
[10]	Irene I. Bouw, Sabine Kampf. Syndrome decoding for Hermite codes with a Sugiyama-type algorithm. Advances in Mathematics of Communications, 2012, 6 (4) : 419-442. doi: 10.3934/amc.2012.6.419
[11]	Anas Chaaban, Vladimir Sidorenko, Christian Senger. On multi-trial Forney-Kovalev decoding of concatenated codes. Advances in Mathematics of Communications, 2014, 8 (1) : 1-20. doi: 10.3934/amc.2014.8.1
[12]	Vladimir Sidorenko, Christian Senger, Martin Bossert, Victor Zyablov. Single-trial decoding of concatenated codes using fixed or adaptive erasing. Advances in Mathematics of Communications, 2010, 4 (1) : 49-60. doi: 10.3934/amc.2010.4.49
[13]	Peter Beelen, Kristian Brander. Efficient list decoding of a class of algebraic-geometry codes. Advances in Mathematics of Communications, 2010, 4 (4) : 485-518. doi: 10.3934/amc.2010.4.485
[14]	Robert F. Bailey, John N. Bray. Decoding the Mathieu group M₁₂. Advances in Mathematics of Communications, 2007, 1 (4) : 477-487. doi: 10.3934/amc.2007.1.477
[15]	Fernando Hernando, Tom Høholdt, Diego Ruano. List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes. Advances in Mathematics of Communications, 2012, 6 (3) : 259-272. doi: 10.3934/amc.2012.6.259
[16]	Henry Cohn, Nadia Heninger. Ideal forms of Coppersmith's theorem and Guruswami-Sudan list decoding. Advances in Mathematics of Communications, 2015, 9 (3) : 311-339. doi: 10.3934/amc.2015.9.311
[17]	Ahmed S. Mansour, Holger Boche, Rafael F. Schaefer. The secrecy capacity of the arbitrarily varying wiretap channel under list decoding. Advances in Mathematics of Communications, 2019, 13 (1) : 11-39. doi: 10.3934/amc.2019002
[18]	Holger Boche, Rafael F. Schaefer. Arbitrarily varying multiple access channels with conferencing encoders: List decoding and finite coordination resources. Advances in Mathematics of Communications, 2016, 10 (2) : 333-354. doi: 10.3934/amc.2016009
[19]	Jonas Eriksson. A weight-based characterization of the set of correctable error patterns under list-of-2 decoding. Advances in Mathematics of Communications, 2007, 1 (3) : 331-356. doi: 10.3934/amc.2007.1.331
[20]	David Clark, Vladimir D. Tonchev. A new class of majority-logic decodable codes derived from polarity designs. Advances in Mathematics of Communications, 2013, 7 (2) : 175-186. doi: 10.3934/amc.2013.7.175

2018 Impact Factor: 0.879

Tools

Metrics

Tools

Article outline

Figures and Tables

2018 Impact Factor: 0.879

Tools

Figures and Tables

2018 Impact Factor: 0.879

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors