Abstract

We apply the eight-state trellis-coded modulation (TCM) using signal constellations of four-dimensional M-ary quadrature-amplitude modulation (4D-MQAM) to optical communication systems for the first time to our knowledge. In the TCM scheme, the free distance of the trellis diagram is equal to the minimum distance between constellation points in partitioned subsets, which enlarges the coding gain effectively. In fact, its asymptotic power efficiency is 3-dB larger than that of the set-partitioned 4D-MQAM (SP-4D-MQAM) format, while their spectral efficiencies are the same. Such theoretical predictions are confirmed through computer simulations on eight-state TCM with constellations of 4D-4QAM (i.e., 4D quadrature phase-shift keying: 4D-QPSK) and 4D-16QAM. In particular, eight-state TCM with 4D-QPSK constellations is practically important because of its simple encoder structure, relatively low computational cost, and high coding gain against dual-polarization QPSK (DP-QPSK) and SP-4D-QPSK. Through measurements of its bit-error rate (BER) performance, we confirm that the coding gain against DP-QPSK is about 3 dB at BER=10−3.

© 2015 Optical Society of America

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References

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  1. K. Kikuchi, “Digital coherent optical communication systems: Fundamentals and future prospects,” IEICE Electronics Express 8, 1642–1662 (2011).
    [Crossref]
  2. S. Ishimura and K. Kikuchi, “Multi-dimensional permutation modulation aiming at both high spectral efficiency and high power efficiency,” in Optical Fiber Communications Conference (OFC 2014), M3A.2 (2014).
    [Crossref]
  3. D. S. Millar, T. Koike-Akino, S. Ö. Ark, K. Kojima, K. Parsons, T. Yoshida, and T. Sugihara, “High-dimensional modulation for coherent optical communications systems,” Opt. Express 22, 8798–8812 (2014).
    [Crossref] [PubMed]
  4. D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, A 24-dimensional modulation format achieving 6 dB asymptotic power efficiency, in Signal Processing in Photonic Communications (SPPCOM 2013), SPM3D.6, (2013).
  5. M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express 17, 10814–10819 (2009).
    [Crossref] [PubMed]
  6. E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. 27, 5115–5126 (2009).
    [Crossref]
  7. L. Coelho and N. Hanik, “Global optimization of fiber-optic communication systems using four-dimensional modulation formats,” in European Conference on Optical Communication (ECOC2011), Mo.2.B (2011).
  8. M. Sjödin, P. Johannisson, and J. Li, “Comparison of 128-SP-QAM with PM-16-QAM,” Opt. Express 20, 8356–8366 (2012).
    [Crossref] [PubMed]
  9. T. A. Eriksson, M. Sjödin, P. A. Andrekson, and M. Karlsson, “Experimental demonstration of 128-SP-QAM in uncompensated longhaul transmission,” in Optical Fiber Communication Conference (OFC2013), OTu3B.2 (2013).
  10. G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory,  IT-28, 55–67 (1982).
    [Crossref]
  11. L. F. Wei, “Trellis-coded modulation with multidimensional constellations,” IEEE Trans. Inform. Theory,  IT-33, 483–501 (1987).
  12. A. R. Calderbank and N. J. A. Sloane, “Four-dimensional modulation with an eight state trellis code,” AT&T Technical J. 64, 1005–1018 (1985).
    [Crossref]
  13. G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Select. Areas Commun. 2, 632–647 (1984).
    [Crossref]
  14. H. Bülow, G. Thielecke, and F. Buchali, “Optical trellis-coded modulation (oTCM),” in Optical Fiber Communication Conference (OFC2004), WM5 (2004).
  15. H. Zhao, E. Agrell, and M. Karlsson, “Trellis-coded modulation in PSK and DPSK communications,” in European Conference on Optical Communication (ECOC2006), We3.P.93 (2006).
  16. T. Sakamoto, A. Chiba, I. Morohashi, and T. Kawanishi, “Optical trellis-coded modulation with multi-parallel MZM,” in European Conference on Optical Communication (ECOC2009), P3.17 (2009).
  17. S. Ishimura and K. Kikuchi, “8-state trellis-coded optical modulation schemes with 4-dimensional set partitioning,” in OptoElectronics and Communication Conference and Australian Conference on Optical Fibre Technology (OECC/ACOFT 2014), TH12B-2 (2014).
  18. Y. Mori, C. Zhang, and K. Kikuchi, “Novel configuration of finite-impulse-response filters tolerant to carrier phase fluctuations in digital coherent optical receivers for higher-order quadrature amplitude modulation signals,” Opt. Express 20, 26236–26251 (2012).
    [Crossref] [PubMed]
  19. Md. S. Faruk, Y. Mori, C. Zhang, and K. Kikuchi, “Proper polarization demultiplexing in coherent optical receiver using constant modulus algorithm with training mode,” in Proceedings of OptoElectronics and Communication Conference (OECC 2010), 9B3-3 (2010).

2014 (1)

2012 (2)

2011 (1)

K. Kikuchi, “Digital coherent optical communication systems: Fundamentals and future prospects,” IEICE Electronics Express 8, 1642–1662 (2011).
[Crossref]

2009 (2)

1987 (1)

L. F. Wei, “Trellis-coded modulation with multidimensional constellations,” IEEE Trans. Inform. Theory,  IT-33, 483–501 (1987).

1985 (1)

A. R. Calderbank and N. J. A. Sloane, “Four-dimensional modulation with an eight state trellis code,” AT&T Technical J. 64, 1005–1018 (1985).
[Crossref]

1984 (1)

G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Select. Areas Commun. 2, 632–647 (1984).
[Crossref]

1982 (1)

G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory,  IT-28, 55–67 (1982).
[Crossref]

Agrell, E.

Andrekson, P. A.

T. A. Eriksson, M. Sjödin, P. A. Andrekson, and M. Karlsson, “Experimental demonstration of 128-SP-QAM in uncompensated longhaul transmission,” in Optical Fiber Communication Conference (OFC2013), OTu3B.2 (2013).

Ark, S. Ö.

Buchali, F.

H. Bülow, G. Thielecke, and F. Buchali, “Optical trellis-coded modulation (oTCM),” in Optical Fiber Communication Conference (OFC2004), WM5 (2004).

Bülow, H.

H. Bülow, G. Thielecke, and F. Buchali, “Optical trellis-coded modulation (oTCM),” in Optical Fiber Communication Conference (OFC2004), WM5 (2004).

Calderbank, A. R.

A. R. Calderbank and N. J. A. Sloane, “Four-dimensional modulation with an eight state trellis code,” AT&T Technical J. 64, 1005–1018 (1985).
[Crossref]

Chiba, A.

T. Sakamoto, A. Chiba, I. Morohashi, and T. Kawanishi, “Optical trellis-coded modulation with multi-parallel MZM,” in European Conference on Optical Communication (ECOC2009), P3.17 (2009).

Coelho, L.

L. Coelho and N. Hanik, “Global optimization of fiber-optic communication systems using four-dimensional modulation formats,” in European Conference on Optical Communication (ECOC2011), Mo.2.B (2011).

Eriksson, T. A.

T. A. Eriksson, M. Sjödin, P. A. Andrekson, and M. Karlsson, “Experimental demonstration of 128-SP-QAM in uncompensated longhaul transmission,” in Optical Fiber Communication Conference (OFC2013), OTu3B.2 (2013).

Faruk, Md. S.

Md. S. Faruk, Y. Mori, C. Zhang, and K. Kikuchi, “Proper polarization demultiplexing in coherent optical receiver using constant modulus algorithm with training mode,” in Proceedings of OptoElectronics and Communication Conference (OECC 2010), 9B3-3 (2010).

Forney, G. D.

G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Select. Areas Commun. 2, 632–647 (1984).
[Crossref]

Gallager, R. G.

G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Select. Areas Commun. 2, 632–647 (1984).
[Crossref]

Hanik, N.

L. Coelho and N. Hanik, “Global optimization of fiber-optic communication systems using four-dimensional modulation formats,” in European Conference on Optical Communication (ECOC2011), Mo.2.B (2011).

Ishimura, S.

S. Ishimura and K. Kikuchi, “8-state trellis-coded optical modulation schemes with 4-dimensional set partitioning,” in OptoElectronics and Communication Conference and Australian Conference on Optical Fibre Technology (OECC/ACOFT 2014), TH12B-2 (2014).

S. Ishimura and K. Kikuchi, “Multi-dimensional permutation modulation aiming at both high spectral efficiency and high power efficiency,” in Optical Fiber Communications Conference (OFC 2014), M3A.2 (2014).
[Crossref]

Johannisson, P.

Karlsson, M.

E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. 27, 5115–5126 (2009).
[Crossref]

M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express 17, 10814–10819 (2009).
[Crossref] [PubMed]

T. A. Eriksson, M. Sjödin, P. A. Andrekson, and M. Karlsson, “Experimental demonstration of 128-SP-QAM in uncompensated longhaul transmission,” in Optical Fiber Communication Conference (OFC2013), OTu3B.2 (2013).

H. Zhao, E. Agrell, and M. Karlsson, “Trellis-coded modulation in PSK and DPSK communications,” in European Conference on Optical Communication (ECOC2006), We3.P.93 (2006).

Kawanishi, T.

T. Sakamoto, A. Chiba, I. Morohashi, and T. Kawanishi, “Optical trellis-coded modulation with multi-parallel MZM,” in European Conference on Optical Communication (ECOC2009), P3.17 (2009).

Kikuchi, K.

Y. Mori, C. Zhang, and K. Kikuchi, “Novel configuration of finite-impulse-response filters tolerant to carrier phase fluctuations in digital coherent optical receivers for higher-order quadrature amplitude modulation signals,” Opt. Express 20, 26236–26251 (2012).
[Crossref] [PubMed]

K. Kikuchi, “Digital coherent optical communication systems: Fundamentals and future prospects,” IEICE Electronics Express 8, 1642–1662 (2011).
[Crossref]

S. Ishimura and K. Kikuchi, “8-state trellis-coded optical modulation schemes with 4-dimensional set partitioning,” in OptoElectronics and Communication Conference and Australian Conference on Optical Fibre Technology (OECC/ACOFT 2014), TH12B-2 (2014).

S. Ishimura and K. Kikuchi, “Multi-dimensional permutation modulation aiming at both high spectral efficiency and high power efficiency,” in Optical Fiber Communications Conference (OFC 2014), M3A.2 (2014).
[Crossref]

Md. S. Faruk, Y. Mori, C. Zhang, and K. Kikuchi, “Proper polarization demultiplexing in coherent optical receiver using constant modulus algorithm with training mode,” in Proceedings of OptoElectronics and Communication Conference (OECC 2010), 9B3-3 (2010).

Koike-Akino, T.

D. S. Millar, T. Koike-Akino, S. Ö. Ark, K. Kojima, K. Parsons, T. Yoshida, and T. Sugihara, “High-dimensional modulation for coherent optical communications systems,” Opt. Express 22, 8798–8812 (2014).
[Crossref] [PubMed]

D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, A 24-dimensional modulation format achieving 6 dB asymptotic power efficiency, in Signal Processing in Photonic Communications (SPPCOM 2013), SPM3D.6, (2013).

Kojima, K.

D. S. Millar, T. Koike-Akino, S. Ö. Ark, K. Kojima, K. Parsons, T. Yoshida, and T. Sugihara, “High-dimensional modulation for coherent optical communications systems,” Opt. Express 22, 8798–8812 (2014).
[Crossref] [PubMed]

D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, A 24-dimensional modulation format achieving 6 dB asymptotic power efficiency, in Signal Processing in Photonic Communications (SPPCOM 2013), SPM3D.6, (2013).

Lang, G. R.

G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Select. Areas Commun. 2, 632–647 (1984).
[Crossref]

Li, J.

Longstaff, F. M.

G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Select. Areas Commun. 2, 632–647 (1984).
[Crossref]

Millar, D. S.

D. S. Millar, T. Koike-Akino, S. Ö. Ark, K. Kojima, K. Parsons, T. Yoshida, and T. Sugihara, “High-dimensional modulation for coherent optical communications systems,” Opt. Express 22, 8798–8812 (2014).
[Crossref] [PubMed]

D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, A 24-dimensional modulation format achieving 6 dB asymptotic power efficiency, in Signal Processing in Photonic Communications (SPPCOM 2013), SPM3D.6, (2013).

Mori, Y.

Y. Mori, C. Zhang, and K. Kikuchi, “Novel configuration of finite-impulse-response filters tolerant to carrier phase fluctuations in digital coherent optical receivers for higher-order quadrature amplitude modulation signals,” Opt. Express 20, 26236–26251 (2012).
[Crossref] [PubMed]

Md. S. Faruk, Y. Mori, C. Zhang, and K. Kikuchi, “Proper polarization demultiplexing in coherent optical receiver using constant modulus algorithm with training mode,” in Proceedings of OptoElectronics and Communication Conference (OECC 2010), 9B3-3 (2010).

Morohashi, I.

T. Sakamoto, A. Chiba, I. Morohashi, and T. Kawanishi, “Optical trellis-coded modulation with multi-parallel MZM,” in European Conference on Optical Communication (ECOC2009), P3.17 (2009).

Parsons, K.

D. S. Millar, T. Koike-Akino, S. Ö. Ark, K. Kojima, K. Parsons, T. Yoshida, and T. Sugihara, “High-dimensional modulation for coherent optical communications systems,” Opt. Express 22, 8798–8812 (2014).
[Crossref] [PubMed]

D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, A 24-dimensional modulation format achieving 6 dB asymptotic power efficiency, in Signal Processing in Photonic Communications (SPPCOM 2013), SPM3D.6, (2013).

Qureshi, S. U.

G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Select. Areas Commun. 2, 632–647 (1984).
[Crossref]

Sakamoto, T.

T. Sakamoto, A. Chiba, I. Morohashi, and T. Kawanishi, “Optical trellis-coded modulation with multi-parallel MZM,” in European Conference on Optical Communication (ECOC2009), P3.17 (2009).

Sjödin, M.

M. Sjödin, P. Johannisson, and J. Li, “Comparison of 128-SP-QAM with PM-16-QAM,” Opt. Express 20, 8356–8366 (2012).
[Crossref] [PubMed]

T. A. Eriksson, M. Sjödin, P. A. Andrekson, and M. Karlsson, “Experimental demonstration of 128-SP-QAM in uncompensated longhaul transmission,” in Optical Fiber Communication Conference (OFC2013), OTu3B.2 (2013).

Sloane, N. J. A.

A. R. Calderbank and N. J. A. Sloane, “Four-dimensional modulation with an eight state trellis code,” AT&T Technical J. 64, 1005–1018 (1985).
[Crossref]

Sugihara, T.

Thielecke, G.

H. Bülow, G. Thielecke, and F. Buchali, “Optical trellis-coded modulation (oTCM),” in Optical Fiber Communication Conference (OFC2004), WM5 (2004).

Ungerboeck, G.

G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory,  IT-28, 55–67 (1982).
[Crossref]

Wei, L. F.

L. F. Wei, “Trellis-coded modulation with multidimensional constellations,” IEEE Trans. Inform. Theory,  IT-33, 483–501 (1987).

Yoshida, T.

Zhang, C.

Y. Mori, C. Zhang, and K. Kikuchi, “Novel configuration of finite-impulse-response filters tolerant to carrier phase fluctuations in digital coherent optical receivers for higher-order quadrature amplitude modulation signals,” Opt. Express 20, 26236–26251 (2012).
[Crossref] [PubMed]

Md. S. Faruk, Y. Mori, C. Zhang, and K. Kikuchi, “Proper polarization demultiplexing in coherent optical receiver using constant modulus algorithm with training mode,” in Proceedings of OptoElectronics and Communication Conference (OECC 2010), 9B3-3 (2010).

Zhao, H.

H. Zhao, E. Agrell, and M. Karlsson, “Trellis-coded modulation in PSK and DPSK communications,” in European Conference on Optical Communication (ECOC2006), We3.P.93 (2006).

AT&T Technical J. (1)

A. R. Calderbank and N. J. A. Sloane, “Four-dimensional modulation with an eight state trellis code,” AT&T Technical J. 64, 1005–1018 (1985).
[Crossref]

IEEE J. Select. Areas Commun. (1)

G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Select. Areas Commun. 2, 632–647 (1984).
[Crossref]

IEEE Trans. Inform. Theory (2)

G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory,  IT-28, 55–67 (1982).
[Crossref]

L. F. Wei, “Trellis-coded modulation with multidimensional constellations,” IEEE Trans. Inform. Theory,  IT-33, 483–501 (1987).

IEICE Electronics Express (1)

K. Kikuchi, “Digital coherent optical communication systems: Fundamentals and future prospects,” IEICE Electronics Express 8, 1642–1662 (2011).
[Crossref]

J. Lightwave Technol. (1)

Opt. Express (4)

Other (9)

Md. S. Faruk, Y. Mori, C. Zhang, and K. Kikuchi, “Proper polarization demultiplexing in coherent optical receiver using constant modulus algorithm with training mode,” in Proceedings of OptoElectronics and Communication Conference (OECC 2010), 9B3-3 (2010).

H. Bülow, G. Thielecke, and F. Buchali, “Optical trellis-coded modulation (oTCM),” in Optical Fiber Communication Conference (OFC2004), WM5 (2004).

H. Zhao, E. Agrell, and M. Karlsson, “Trellis-coded modulation in PSK and DPSK communications,” in European Conference on Optical Communication (ECOC2006), We3.P.93 (2006).

T. Sakamoto, A. Chiba, I. Morohashi, and T. Kawanishi, “Optical trellis-coded modulation with multi-parallel MZM,” in European Conference on Optical Communication (ECOC2009), P3.17 (2009).

S. Ishimura and K. Kikuchi, “8-state trellis-coded optical modulation schemes with 4-dimensional set partitioning,” in OptoElectronics and Communication Conference and Australian Conference on Optical Fibre Technology (OECC/ACOFT 2014), TH12B-2 (2014).

D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, A 24-dimensional modulation format achieving 6 dB asymptotic power efficiency, in Signal Processing in Photonic Communications (SPPCOM 2013), SPM3D.6, (2013).

S. Ishimura and K. Kikuchi, “Multi-dimensional permutation modulation aiming at both high spectral efficiency and high power efficiency,” in Optical Fiber Communications Conference (OFC 2014), M3A.2 (2014).
[Crossref]

T. A. Eriksson, M. Sjödin, P. A. Andrekson, and M. Karlsson, “Experimental demonstration of 128-SP-QAM in uncompensated longhaul transmission,” in Optical Fiber Communication Conference (OFC2013), OTu3B.2 (2013).

L. Coelho and N. Hanik, “Global optimization of fiber-optic communication systems using four-dimensional modulation formats,” in European Conference on Optical Communication (ECOC2011), Mo.2.B (2011).

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Figures (7)

Fig. 1
Fig. 1 Set-partitioning process for 4D-MQAM constellations. First-step partitioning generates subsets R0 and R1, and second-step partitioning does subsets S0, ···, S7. Each set-partitioning process doubles MSED between constellation points.
Fig. 2
Fig. 2 Encoder structure for eight-state TCM. Bits b1, b2, and b3 are coded by the convolutional process and other input bits b4, ··· are not coded.
Fig. 3
Fig. 3 Trellis diagram of the code generated by the convolutional encoder shown in Fig. 2. Dots represent the shift-register state of the convolutional encoder. The closest two paths are shown in red lines. Three output bits b1, b2, b3 are shown on each path. The number of selected subsets is also shown on each path.
Fig. 4
Fig. 4 SE as a function of 1/γ for DP-MQAM, M2/2-SP-QAM and eight-state TCM with 4D-MQAM constellations. Two-dimensional QAM orders are M = 4, 16, and 64.
Fig. 5
Fig. 5 BERs as a function of Eb/N0. (a): Eight-state TCM with 4D-QPSK constellations, uncoded DP-QPSK, and uncoded PS-QPSK. (b): Eight-state TCM with 4D-16QAM constellations, uncoded DP-16QAM, and uncoded 128-SP-QAM.
Fig. 6
Fig. 6 Experimental set up for the BER measurement of eight-state TCM with 4D-QPSK constellations. (a): Experimental system configuration. (b): DSP at the transmitter. (c): DSP at the receiver.
Fig. 7
Fig. 7 BERs of eight-state TCM with 4D-QPSK constellations (red color) and DP-QPSK (black color) at 10 Gbaud as a function of Eb/N0. Curves, dots, and triangles represent simulation results, back-to-back BERs, and BERs after 50-km transmission, respectively.

Tables (1)

Tables Icon

Table 1 Assignment of three output bits to the partitioned subset.

Equations (10)

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

E = [ E I x E Q x E I y E Q y ] ,
E 1 ± = [ ± 1 0 0 0 ] , E 2 ± = [ 0 ± 1 0 0 ] , E 3 ± = [ 0 0 ± 1 0 ] , E 4 ± = [ 0 0 0 ± 1 ] .
E 1 = d 0 2 [ + 1 + 1 + 1 + 1 ] , E 2 = d 0 2 [ + 1 + 1 1 1 ] , E 3 = d 0 2 [ + 1 1 + 1 1 ] , E 4 = d 0 2 [ + 1 1 1 + 1 ] , E 5 = d 0 2 [ 1 1 1 1 ] , E 6 = d 0 2 [ 1 1 + 1 + 1 ] , E 7 = d 0 2 [ 1 + 1 1 + 1 ] , E 8 = d 0 2 [ 1 + 1 + 1 1 ] , E 9 = d 0 2 [ + 1 + 1 + 1 1 ] , E 10 = d 0 2 [ + 1 + 1 1 + 1 ] , E 11 = d 0 2 [ + 1 1 + 1 + 1 ] , E 12 = d 0 2 [ 1 + 1 + 1 + 1 ] , E 13 = d 0 2 [ 1 1 1 + 1 ] , E 14 = d 0 2 [ 1 1 + 1 1 ] , E 15 = d 0 2 [ 1 + 1 1 1 ] , E 16 = d 0 2 [ + 1 1 1 1 ] .
ε free 2 = d [ S 0 , S 2 ] 2 + d [ S 0 , S 1 ] 2 = 2 d 0 2 + 2 d 0 2 = 4 d 0 2 ,
ε min 2 = 4 d 0 2 .
d min 2 = min [ ε free 2 , ε min 2 ] .
γ = d min 2 4 E b .
SE = log 2 N 2 ,
γ = 3 ( 2 log 2 M 1 ) M 1 ,
SE = log 2 M 1 2 .

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