Abstract

Large-girth quasi-cyclic LDPC codes have been experimentally evaluated for use in PMD compensation by turbo equalization for a 10 Gb/s NRZ optical transmission system, and observing one sample per bit. Net effective coding gain improvement for girth-10, rate 0.906 code of length 11936 over maximum a posteriori probability (MAP) detector for differential group delay of 125 ps is 6.25 dB at BER of 10-6. Girth-10 LDPC code of rate 0.8 outperforms the girth-10 code of rate 0.906 by 2.75 dB, and provides the net effective coding gain improvement of 9 dB at the same BER. It is experimentally determined that girth-10 LDPC codes of length around 15000 approach channel capacity limit within 1.25 dB.

© 2008 Optical Society of America

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Simultaneous chromatic dispersion and PMD compensation by using coded-OFDM and girth-10 LDPC codes

Ivan B. Djordjevic, Lei Xu, and Ting Wang
Opt. Express 16(14) 10269-10278 (2008)

References

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  1. T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
    [Crossref]
  2. I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. 19, 1163–1165 (2007).
    [Crossref]
  3. B. Vasic, I. B. Djordjevic, and R. Kostuk, “Low-density parity check codes and iterative decoding for long-haul optical communication systems,” J. Lightwave Technol. 21, 438–446 (2003).
    [Crossref]
  4. I. B. Djordjevic, S. Sankaranarayanan, S. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40-Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
    [Crossref]
  5. M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory 50, 1788–1793 (2004).
    [Crossref]
  6. O. Milenkovic, I. B. Djordjevic, and B. Vasic, “Block-circulant low-density parity-check codes for optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 294–299 (2004).
    [Crossref]
  7. I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. JWA53.
  8. R. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inf. Theory IT-27, 533–547 (1981).
    [Crossref]
  9. L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. 19, 1852–1854 (2007).
    [Crossref]
  10. H. Xiao-Yu et al., “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE Globecom 2, 1036–1036E (2001).
  11. Y. Miyata, W. Matsumoto, H. Yoshida, and T. Mizuochi, “Efficient FEC for optical communications using concatenated codes to combat error-floor,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. OTuE4.
  12. M. Arabaci, I. B. Djordjevic, and R. Saunders, “FPGA implementation of LDPC decoders for large girth LDPC codes,” in preparation.

2007 (2)

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. 19, 1163–1165 (2007).
[Crossref]

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. 19, 1852–1854 (2007).
[Crossref]

2006 (1)

I. B. Djordjevic, S. Sankaranarayanan, S. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40-Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

2004 (3)

M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory 50, 1788–1793 (2004).
[Crossref]

O. Milenkovic, I. B. Djordjevic, and B. Vasic, “Block-circulant low-density parity-check codes for optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 294–299 (2004).
[Crossref]

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

2003 (1)

2001 (1)

H. Xiao-Yu et al., “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE Globecom 2, 1036–1036E (2001).

1981 (1)

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

Akita, M.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Arabaci, M.

M. Arabaci, I. B. Djordjevic, and R. Saunders, “FPGA implementation of LDPC decoders for large girth LDPC codes,” in preparation.

Batshon, H. G.

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. 19, 1852–1854 (2007).
[Crossref]

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. 19, 1163–1165 (2007).
[Crossref]

Chilappagari, S.

I. B. Djordjevic, S. Sankaranarayanan, S. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40-Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

Cvijetic, M.

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. 19, 1163–1165 (2007).
[Crossref]

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. 19, 1852–1854 (2007).
[Crossref]

I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. JWA53.

Djordjevic, I. B.

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. 19, 1163–1165 (2007).
[Crossref]

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. 19, 1852–1854 (2007).
[Crossref]

I. B. Djordjevic, S. Sankaranarayanan, S. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40-Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

O. Milenkovic, I. B. Djordjevic, and B. Vasic, “Block-circulant low-density parity-check codes for optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 294–299 (2004).
[Crossref]

B. Vasic, I. B. Djordjevic, and R. Kostuk, “Low-density parity check codes and iterative decoding for long-haul optical communication systems,” J. Lightwave Technol. 21, 438–446 (2003).
[Crossref]

I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. JWA53.

M. Arabaci, I. B. Djordjevic, and R. Saunders, “FPGA implementation of LDPC decoders for large girth LDPC codes,” in preparation.

Fossorier, M. P. C.

M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory 50, 1788–1793 (2004).
[Crossref]

Fujita, H.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Kobayashi, T.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Kostuk, R.

Kubo, K.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Kueppers, F.

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. 19, 1852–1854 (2007).
[Crossref]

Kuno, K.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Matsumoto, W.

Y. Miyata, W. Matsumoto, H. Yoshida, and T. Mizuochi, “Efficient FEC for optical communications using concatenated codes to combat error-floor,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. OTuE4.

Milenkovic, O.

O. Milenkovic, I. B. Djordjevic, and B. Vasic, “Block-circulant low-density parity-check codes for optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 294–299 (2004).
[Crossref]

Minkov, L. L.

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. 19, 1852–1854 (2007).
[Crossref]

Miyata, Y.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Y. Miyata, W. Matsumoto, H. Yoshida, and T. Mizuochi, “Efficient FEC for optical communications using concatenated codes to combat error-floor,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. OTuE4.

Mizuochi, T.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Y. Miyata, W. Matsumoto, H. Yoshida, and T. Mizuochi, “Efficient FEC for optical communications using concatenated codes to combat error-floor,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. OTuE4.

Motoshima, K.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Ouchi, K.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Sankaranarayanan, S.

I. B. Djordjevic, S. Sankaranarayanan, S. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40-Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

Saunders, R.

M. Arabaci, I. B. Djordjevic, and R. Saunders, “FPGA implementation of LDPC decoders for large girth LDPC codes,” in preparation.

Shimizu, K.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Tagami, H.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Tanner, R.

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

Vasic, B.

I. B. Djordjevic, S. Sankaranarayanan, S. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40-Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

O. Milenkovic, I. B. Djordjevic, and B. Vasic, “Block-circulant low-density parity-check codes for optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 294–299 (2004).
[Crossref]

B. Vasic, I. B. Djordjevic, and R. Kostuk, “Low-density parity check codes and iterative decoding for long-haul optical communication systems,” J. Lightwave Technol. 21, 438–446 (2003).
[Crossref]

Wang, T.

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. 19, 1163–1165 (2007).
[Crossref]

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. 19, 1852–1854 (2007).
[Crossref]

I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. JWA53.

Xiao-Yu, H.

H. Xiao-Yu et al., “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE Globecom 2, 1036–1036E (2001).

Xu, L.

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. 19, 1852–1854 (2007).
[Crossref]

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. 19, 1163–1165 (2007).
[Crossref]

I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. JWA53.

Yoshida, H.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

Y. Miyata, W. Matsumoto, H. Yoshida, and T. Mizuochi, “Efficient FEC for optical communications using concatenated codes to combat error-floor,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. OTuE4.

IEEE J. Sel. Top. Quantum Electron. (3)

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386 (2004).
[Crossref]

I. B. Djordjevic, S. Sankaranarayanan, S. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40-Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

O. Milenkovic, I. B. Djordjevic, and B. Vasic, “Block-circulant low-density parity-check codes for optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 294–299 (2004).
[Crossref]

IEEE Photon. Technol. Lett. (2)

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC coding based turbo equalization,” IEEE Photon. Technol. Lett. 19, 1163–1165 (2007).
[Crossref]

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. 19, 1852–1854 (2007).
[Crossref]

IEEE Trans. Inf. Theory (2)

M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory 50, 1788–1793 (2004).
[Crossref]

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

J. Lightwave Technol. (1)

Proc. IEEE Globecom (1)

H. Xiao-Yu et al., “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE Globecom 2, 1036–1036E (2001).

Other (3)

Y. Miyata, W. Matsumoto, H. Yoshida, and T. Mizuochi, “Efficient FEC for optical communications using concatenated codes to combat error-floor,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. OTuE4.

M. Arabaci, I. B. Djordjevic, and R. Saunders, “FPGA implementation of LDPC decoders for large girth LDPC codes,” in preparation.

I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 24–28 2008, Paper no. JWA53.

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Figures (4)

Fig. 1.
Fig. 1.

Experimental setup

Fig. 2.
Fig. 2.

(a) trellis description (2m+1=5), (b) measured conditional PDFs for different DGD values.

Fig. 3.
Fig. 3.

Channel capacity in the presence of PMD (for an NRZ transmission system operating at 10-Gb/s).

Fig. 4.
Fig. 4.

(a) Performance evaluation of LDPC codes in the presence of constant amount of DGD (125ps). (b) Performance evaluation of LDPC(11936,10819) code of rate R=0.906, for different values of DGD.

Tables (1)

Tables Icon

Table 1. Tested LDPC codes

Equations (2)

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

H = [ I I I I I D S [ 1 ] D S [ 2 ] D S [ c 1 ] I D 2 S [ 1 ] D 2 S [ 2 ] D 2 S [ c 1 ] I D ( r 1 ) S [ 1 ] D ( r 1 ) S [ 2 ] D ( r 1 ) S [ c 1 ] ]
d { 1 + w w 2 ( ( w 1 ) ( g 2 ) 4 1 ) , g 2 = 2 m + 1 1 + w w 2 ( ( w 1 ) ( g 2 ) 4 1 ) + ( w 1 ) ( g 2 ) 4 , g 2 = 2 m

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