Abstract

A class of generalized low-density parity-check (GLDPC) codes suitable for optical communications is proposed, which consists of multiple local codes. It is shown that Hamming, BCH, and Reed-Muller codes can be used as local codes, and that the maximum a posteriori probability (MAP) decoding of these local codes by Ashikhmin-Lytsin algorithm is feasible in terms of complexity and performance. We demonstrate that record coding gains can be obtained from properly designed GLDPC codes, derived from multiple component codes. We then show that several recently proposed classes of LDPC codes such as convolutional and spatially-coupled codes can be described using the concept of GLDPC coding, which indicates that the GLDPC coding can be used as a unified platform for advanced FEC enabling ultra-high speed optical transport. The proposed class of GLDPC codes is also suitable for code-rate adaption, to adjust the error correction strength depending on the optical channel conditions.

© 2014 Optical Society of America

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References

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  1. M. Cvijetic and I. B. Djordjevic, Advanced Optical Communication Systems and Networks (Artech House, 2013).
  2. I. B. Djordjevic, L. Xu, and T. Wang, “On the reduced-complexity of LDPC decoders for ultra-high-speed optical transmission,” Opt. Express 18(22), 23371–23377 (2010).
    [Crossref] [PubMed]
  3. I. B. Djordjevic, M. Arabaci, and L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightwave Technol. 27(16), 3518–3530 (2009).
    [Crossref]
  4. D. Chang, F. Yu, Z. Xiao, N. Stojanovic, F. N. Hauske, Y. Cai, C. Xie, L. Li, X. Xu, and Q. Xiong, “LDPC convolutional codes using layered decoding algorithm for high speed coherent optical transmission,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper OW1H.4.
    [Crossref]
  5. B. P. Smith, A. Farhood, A. Hunt, F. R. Kschischang, and J. Lodge, “Staircase codes: FEC for 100 Gb/s OTN,” J. Lightwave Technol. 30(1), 110–117 (2012).
    [Crossref]
  6. Y. Zhang and I. B. Djordjevic, “Staircase rate-adaptive LDPC-coded modulation for high-speed intelligent optical transmission,” in Proc. OFC, 9–13 March 2014, San Francisco, California, USA (2014), Paper M3A.6.
    [Crossref]
  7. K. Sugihara, Y. Miyata, T. Sugihara, K. Kubo, H. Yoshida, W. Matsumoto, and T. Mizuochi, “A Spatially-coupled type LDPC code with an NCG of 12 dB for optical transmission beyond 100 Gb/s,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper OM2B.4.
    [Crossref]
  8. I. B. Djordjevic, “Advances in error correction coding for high-speed optical transmission,” in Proc. IEEE Photonics Conference (IPC 2013), 8–12 September 2013, Hyatt Regency Bellevue, Washington, USA (Invited paper) (2013), Paper MG3.1.
    [Crossref]
  9. I. B. Djordjevic and T. Wang, “Rate-Adaptive irregular QC-LDPC codes from pairwise balanced designs for ultra-high-speed optical transport,” in CLEO 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper CM1G.8.
  10. I. Anderson, Combinatorial Designs and Tournaments (Oxford University, 1997).
  11. R. M. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inf. Theory 27(5), 533–547 (1981).
    [Crossref]
  12. I. B. Djordjevic, O. Milenkovic, and B. Vasic, “Generalized low-density parity-check codes for optical communication systems,” J. Lightwave Technol. 23(5), 1939–1946 (2005).
    [Crossref]
  13. I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “GLDPC codes with Reed-Muller component codes suitable for optical communications,” IEEE Commun. Lett. 12(9), 684–686 (2008).
    [Crossref]
  14. J. Boutros, O. Pothier, and G. Zemor, “Generalized low density (Tanner) codes,” in Proc. IEEE Int. Conf. Comm. (ICC’99) (1999), pp. 441–445.
    [Crossref]
  15. M. Lentmaier and K. Sh. Zigangirov, “On generalized low-density parity-check codes based on Hamming component codes,” IEEE Commun. Lett. 3(8), 248–250 (1999).
    [Crossref]
  16. V. A. Zyablov, R. Johannesson, and M. Loncar, “Low-complexity error correction of Hamming-code-based LDPC codes,” Probl. Inf. Transm. 45(2), 95–109 (2009).
    [Crossref]
  17. Y. Chen and K. K. Parhi, “Parallel decoding of interleaved single parity check turbo product codes,” in Proc. IEEE Workshop on Signal Processing Systems 2002 (SIPS'02), pp. 27 – 32, 16–18 Oct. 2002.
  18. W. E. Ryan and S. Lin, Channel Codes: Classical and Modern (Cambridge University Press, 2009).
  19. I. B. Djordjevic, “Spatial-domain-based hybrid multidimensional coded-modulation schemes enabling multi-Tb/s optical transport,” J. Lightwave Technol. 30(14), 2315–2328 (2012).
    [Crossref]
  20. Y. Zhao, J. Qi, F. N. Hauske, C. Xie, D. Pflueger, and G. Bauch, “Beyond 100G optical channel noise modeling for optimized soft-decision FEC performance,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2012, OSA Technical Digest (online) (Optical Society of America, 2012), paper OW1H.3.
    [Crossref]
  21. D. E. Hocevar, “A reduced complexity decoder architecture via layered decoding of LDPC codes,” in Proc. IEEE Workshop on Signal Processing Systems 2004 (SIPS 2004), 13–15 Oct. (2004), pp. 107–112.
    [Crossref]
  22. G. Liva, W. E. Ryan, and M. Chiani, “Design of quasi-cyclic Tanner codes with low error floors,” in 4th International Symposium on Turbo Codes (ISTC-2006), April 3-7, Munich, Germany, (2006).
  23. S. Abu-Surra, G. Liva, and W. E. Ryan, “Low-floor Tanner codes via Hamming-node or RSCC-node doping,” in Proc. 16th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2006) (2006), Vol. 3857, pp. 245–254.
  24. N. Miladinovic and M. Fossorier, “Generalized LDPC codes with Reed-Solomon and BCH codes as component codes for binary channels,” in Proc. IEEE Global Telecommunications Conference (GLOBECOM ’05) (2005), 1239–1244.
  25. A. Ashikhmin and S. Litsyn, “Simple MAP decoding of first-order Reed-Muller and Hamming codes,” IEEE Trans. Inf. Theory 50(8), 1812–1818 (2004).
    [Crossref]

2012 (2)

2010 (1)

2009 (2)

I. B. Djordjevic, M. Arabaci, and L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightwave Technol. 27(16), 3518–3530 (2009).
[Crossref]

V. A. Zyablov, R. Johannesson, and M. Loncar, “Low-complexity error correction of Hamming-code-based LDPC codes,” Probl. Inf. Transm. 45(2), 95–109 (2009).
[Crossref]

2008 (1)

I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “GLDPC codes with Reed-Muller component codes suitable for optical communications,” IEEE Commun. Lett. 12(9), 684–686 (2008).
[Crossref]

2005 (1)

2004 (1)

A. Ashikhmin and S. Litsyn, “Simple MAP decoding of first-order Reed-Muller and Hamming codes,” IEEE Trans. Inf. Theory 50(8), 1812–1818 (2004).
[Crossref]

1999 (1)

M. Lentmaier and K. Sh. Zigangirov, “On generalized low-density parity-check codes based on Hamming component codes,” IEEE Commun. Lett. 3(8), 248–250 (1999).
[Crossref]

1981 (1)

R. M. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inf. Theory 27(5), 533–547 (1981).
[Crossref]

Arabaci, M.

Ashikhmin, A.

A. Ashikhmin and S. Litsyn, “Simple MAP decoding of first-order Reed-Muller and Hamming codes,” IEEE Trans. Inf. Theory 50(8), 1812–1818 (2004).
[Crossref]

Cvijetic, M.

I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “GLDPC codes with Reed-Muller component codes suitable for optical communications,” IEEE Commun. Lett. 12(9), 684–686 (2008).
[Crossref]

Djordjevic, I. B.

Farhood, A.

Fossorier, M.

N. Miladinovic and M. Fossorier, “Generalized LDPC codes with Reed-Solomon and BCH codes as component codes for binary channels,” in Proc. IEEE Global Telecommunications Conference (GLOBECOM ’05) (2005), 1239–1244.

Hunt, A.

Johannesson, R.

V. A. Zyablov, R. Johannesson, and M. Loncar, “Low-complexity error correction of Hamming-code-based LDPC codes,” Probl. Inf. Transm. 45(2), 95–109 (2009).
[Crossref]

Kschischang, F. R.

Lentmaier, M.

M. Lentmaier and K. Sh. Zigangirov, “On generalized low-density parity-check codes based on Hamming component codes,” IEEE Commun. Lett. 3(8), 248–250 (1999).
[Crossref]

Litsyn, S.

A. Ashikhmin and S. Litsyn, “Simple MAP decoding of first-order Reed-Muller and Hamming codes,” IEEE Trans. Inf. Theory 50(8), 1812–1818 (2004).
[Crossref]

Lodge, J.

Loncar, M.

V. A. Zyablov, R. Johannesson, and M. Loncar, “Low-complexity error correction of Hamming-code-based LDPC codes,” Probl. Inf. Transm. 45(2), 95–109 (2009).
[Crossref]

Miladinovic, N.

N. Miladinovic and M. Fossorier, “Generalized LDPC codes with Reed-Solomon and BCH codes as component codes for binary channels,” in Proc. IEEE Global Telecommunications Conference (GLOBECOM ’05) (2005), 1239–1244.

Milenkovic, O.

Minkov, L.

Smith, B. P.

Tanner, R. M.

R. M. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inf. Theory 27(5), 533–547 (1981).
[Crossref]

Vasic, B.

Wang, T.

I. B. Djordjevic, L. Xu, and T. Wang, “On the reduced-complexity of LDPC decoders for ultra-high-speed optical transmission,” Opt. Express 18(22), 23371–23377 (2010).
[Crossref] [PubMed]

I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “GLDPC codes with Reed-Muller component codes suitable for optical communications,” IEEE Commun. Lett. 12(9), 684–686 (2008).
[Crossref]

Xu, L.

I. B. Djordjevic, L. Xu, and T. Wang, “On the reduced-complexity of LDPC decoders for ultra-high-speed optical transmission,” Opt. Express 18(22), 23371–23377 (2010).
[Crossref] [PubMed]

I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “GLDPC codes with Reed-Muller component codes suitable for optical communications,” IEEE Commun. Lett. 12(9), 684–686 (2008).
[Crossref]

Zigangirov, K. Sh.

M. Lentmaier and K. Sh. Zigangirov, “On generalized low-density parity-check codes based on Hamming component codes,” IEEE Commun. Lett. 3(8), 248–250 (1999).
[Crossref]

Zyablov, V. A.

V. A. Zyablov, R. Johannesson, and M. Loncar, “Low-complexity error correction of Hamming-code-based LDPC codes,” Probl. Inf. Transm. 45(2), 95–109 (2009).
[Crossref]

IEEE Commun. Lett. (2)

I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “GLDPC codes with Reed-Muller component codes suitable for optical communications,” IEEE Commun. Lett. 12(9), 684–686 (2008).
[Crossref]

M. Lentmaier and K. Sh. Zigangirov, “On generalized low-density parity-check codes based on Hamming component codes,” IEEE Commun. Lett. 3(8), 248–250 (1999).
[Crossref]

IEEE Trans. Inf. Theory (2)

R. M. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inf. Theory 27(5), 533–547 (1981).
[Crossref]

A. Ashikhmin and S. Litsyn, “Simple MAP decoding of first-order Reed-Muller and Hamming codes,” IEEE Trans. Inf. Theory 50(8), 1812–1818 (2004).
[Crossref]

J. Lightwave Technol. (4)

Opt. Express (1)

Probl. Inf. Transm. (1)

V. A. Zyablov, R. Johannesson, and M. Loncar, “Low-complexity error correction of Hamming-code-based LDPC codes,” Probl. Inf. Transm. 45(2), 95–109 (2009).
[Crossref]

Other (15)

Y. Chen and K. K. Parhi, “Parallel decoding of interleaved single parity check turbo product codes,” in Proc. IEEE Workshop on Signal Processing Systems 2002 (SIPS'02), pp. 27 – 32, 16–18 Oct. 2002.

W. E. Ryan and S. Lin, Channel Codes: Classical and Modern (Cambridge University Press, 2009).

J. Boutros, O. Pothier, and G. Zemor, “Generalized low density (Tanner) codes,” in Proc. IEEE Int. Conf. Comm. (ICC’99) (1999), pp. 441–445.
[Crossref]

M. Cvijetic and I. B. Djordjevic, Advanced Optical Communication Systems and Networks (Artech House, 2013).

D. Chang, F. Yu, Z. Xiao, N. Stojanovic, F. N. Hauske, Y. Cai, C. Xie, L. Li, X. Xu, and Q. Xiong, “LDPC convolutional codes using layered decoding algorithm for high speed coherent optical transmission,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper OW1H.4.
[Crossref]

Y. Zhang and I. B. Djordjevic, “Staircase rate-adaptive LDPC-coded modulation for high-speed intelligent optical transmission,” in Proc. OFC, 9–13 March 2014, San Francisco, California, USA (2014), Paper M3A.6.
[Crossref]

K. Sugihara, Y. Miyata, T. Sugihara, K. Kubo, H. Yoshida, W. Matsumoto, and T. Mizuochi, “A Spatially-coupled type LDPC code with an NCG of 12 dB for optical transmission beyond 100 Gb/s,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper OM2B.4.
[Crossref]

I. B. Djordjevic, “Advances in error correction coding for high-speed optical transmission,” in Proc. IEEE Photonics Conference (IPC 2013), 8–12 September 2013, Hyatt Regency Bellevue, Washington, USA (Invited paper) (2013), Paper MG3.1.
[Crossref]

I. B. Djordjevic and T. Wang, “Rate-Adaptive irregular QC-LDPC codes from pairwise balanced designs for ultra-high-speed optical transport,” in CLEO 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper CM1G.8.

I. Anderson, Combinatorial Designs and Tournaments (Oxford University, 1997).

Y. Zhao, J. Qi, F. N. Hauske, C. Xie, D. Pflueger, and G. Bauch, “Beyond 100G optical channel noise modeling for optimized soft-decision FEC performance,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2012, OSA Technical Digest (online) (Optical Society of America, 2012), paper OW1H.3.
[Crossref]

D. E. Hocevar, “A reduced complexity decoder architecture via layered decoding of LDPC codes,” in Proc. IEEE Workshop on Signal Processing Systems 2004 (SIPS 2004), 13–15 Oct. (2004), pp. 107–112.
[Crossref]

G. Liva, W. E. Ryan, and M. Chiani, “Design of quasi-cyclic Tanner codes with low error floors,” in 4th International Symposium on Turbo Codes (ISTC-2006), April 3-7, Munich, Germany, (2006).

S. Abu-Surra, G. Liva, and W. E. Ryan, “Low-floor Tanner codes via Hamming-node or RSCC-node doping,” in Proc. 16th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2006) (2006), Vol. 3857, pp. 245–254.

N. Miladinovic and M. Fossorier, “Generalized LDPC codes with Reed-Solomon and BCH codes as component codes for binary channels,” in Proc. IEEE Global Telecommunications Conference (GLOBECOM ’05) (2005), 1239–1244.

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Figures (5)

Fig. 1
Fig. 1 The graph description of length-3544 GLDPC code: (a) RM(1,3) component code, (b) single-parity check code, (c) the parity-check matrix of girth-10, column-weight-3, global (3544,2215) code suitable for parallelization. The variable nodes in local codes are denoted with circles, while the parity-check nodes with squares. Row indices used in Fig. 1(c) for global code are 1, 2 and 7. The cardinality of set S is 8, and it is given by {0, 1, 3,14,22,31,60,121}. The bit nodes of the global code within the block-column are grouped into a block-processing element (BPE), while the parity-checks within block-row are grouped into a super-node processing element (SNPE). BPEs and SNPEs operate in parallel, while the processing within BPEs/SNPEs is in a serial fashion.
Fig. 2
Fig. 2 BER performance of proposed GLDPC (16935, K)-coded PDM QPSK. The information symbol rate is set to 25 GS/s (with aggregate information bit rate of 100 Gb/s).
Fig. 3
Fig. 3 BER performance of proposed GLDPC (53168, K)-coded PDM QPSK. The information symbol rate is set to 25 GS/s (with aggregate information bit rate of 100 Gb/s).
Fig. 4
Fig. 4 BER performance of proposed GLDPC (59993, 49397)-coded PDM QPSK. The information symbol rate is set to 25 GS/s (with aggregate information bit rate of 100 Gb/s).
Fig. 5
Fig. 5 Shannon limits corresponding to code rates used in Figs. 24.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

1 W n W d ( 1 k+1 n )R R G ,
CR= ( N LDPC K LDPC ) w r + N LDPC ( 2 w c +1 ) ( N/d ) i n i log 2 n i +( NN/d )×3 × t LDPC t GLDPC ,
μ v ( 0 ) =log[ log( Pr( v=0| y v ) Pr( v=1| y v ) ) ],
λ v,s ( i ) = μ v ( 0 ) + s's Λ s',v ( i ) .
μ v ( i ) = μ v ( 0 ) + s Λ s,v ( i ) .
x ^ v ={ 1,if μ v ( j ) <0 0,otherwise .
CR=[( 1693513550 )×15+16935×(2×3+1)]×30 / [( 16935/4000 )×15 log 2 15+( 1693516935/4000 )×3]×60 =1.6594
CR=[( 5316843201 )×16+53168×( 3+4 )]×30/{[( 53168/10000 )×( 16 log 2 16 )+(53168 53168/10000)×3]×60}=1.6632
Subcodes 1 to L H G = [ 1 1 1 1 Bit nodes 1to n 1 1 1 1 Bit nodes with overlap 1 1 1 1 1 1 1 1 1 Bit nodes correponding to the L-th local code ]
H=[ H B T π 1 ( H B ) T π W1 ( H B ) T ],
H=[ I I I ... I I P S[ 1 ] P S[ 2 ] ... P S[ c1 ] I P 2S[ 1 ] P 2S[ 2 ] ... P 2S[ c1 ] ... ... ... ... ... I P ( r1 )S[ 1 ] P ( r1 )S[ 2 ] ... P ( r1 )S[ c1 ] ],
H t =[ I I I I I I P S[ 1 ] P S[ 2 ] P S[ 3 ] P S[ 4 ] I P 2S[ 1 ] P 2S[ 2 ] P 2S[ 3 ] P 2S[ 4 ] ],
H=[ I P S[ 1 ] P S[ 2 ] P 2S[ 1 ] P 2S[ 2 ] P 2S[ 3 ] I I I P S[ 3 ] P S[ 1 ] P S[ 2 ] P 2S[ 1 ] P 2S[ 2 ] P 2S[ 3 ] I I P S[ 3 ] ],
H=[ I I P S[ 1 ] P S[ 2 ] I I P 2S[ 1 ] P 2S[ 2 ] P S[ 3 ] P S[ 4 ] P 2S[ 3 ] P 2S[ 4 ] ]
H G =[ 1 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 ].
H=[ I I I I I P S[ 1 ] P S[ 2 ] P S[ 3 ] I P 2S[ 1 ] P 2S[ 2 ] P 2S[ 3 ] ].

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