Expanding the compute-and-forward framework: Unequal powers, signal levels, and multiple linear combinations

IEEE Transactions on Information Theory

Volumn 62, Issue 9, 2016, Pages 4879-4909

  Nazer, Bobak ^a   Cadambe, Viveck R ^b   Ntranos, Vasilis ^c   Caire, Giuseppe ^c,d  

^a Boston University   (United States)

^b The Pennsylvania State University   (United States)

^c University of Southern California   (United States)

^d TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

compute and forward; Interference; nested lattice codes; successive decoding; unequal powers

Indexed keywords

COMPUTER APPLICATIONS; INFORMATION THEORY; WAVE INTERFERENCE;

COMPUTE AND FORWARDS; END-TO-END COMMUNICATION; GAUSSIAN NETWORKS; INTERFERENCE ALIGNMENT; LINEAR COMBINATIONS; NESTED LATTICE CODES; SUCCESSIVE DECODING; UNEQUAL POWERS;

DECODING;

EID: 84983514839     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2016.2593633     Document Type: Article

References (92)

1. B. Nazer and M. Gastpar, "Compute-and-forward: Harnessing interference through structured codes," IEEE Trans. Inf. Theory, vol. 57, no. 10, pp. 6463-6486, Oct. 2011.
2. B. Nazer and M. Gastpar, "Reliable physical layer network coding," Proc. IEEE, vol. 99, no. 3, pp. 438-460, Mar. 2011.
3. J. Zhan, B. Nazer, U. Erez, and M. Gastpar, "Integer-forcing linear receivers," IEEE Trans. Inf. Theory, vol. 60, no. 12, pp. 7661-7685, Dec. 2014.
4. S.-N. Hong and G. Caire, "Reverse compute and forward: A lowcomplexity architecture for downlink distributed antenna systems," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Cambridge, MA, USA, Jul. 2012, pp. 1147-1151.
5. S.-N. Hong and G. Caire, "Compute-and-forward strategies for cooperative distributed antenna systems," IEEE Trans. Inf. Theory, vol. 59, no. 9, pp. 5227-5243, Sep. 2013.
6. A. Sakzad, J. Harshan, and E. Viterbo, "Integer-forcing MIMO linear receivers based on lattice reduction," IEEE Trans. Wireless Commun., vol. 12, no. 10, pp. 4905-4915, Oct. 2013.
7. O. Ordentlich, U. Erez, and B. Nazer, "Successive integer-forcing and its sum-rate optimality," in Proc. 51st Annu. Allerton Conf. Commun., Control, Comput., Monticello, IL, USA, Oct. 2013, pp. 282-292.
8. O. Ordentlich and U. Erez, "Precoded integer-forcing universally achieves the MIMO capacity to within a constant gap," IEEE Trans. Inf. Theory, vol. 61, no. 1, pp. 323-340, Jan. 2015.
9. O. Ordentlich, U. Erez, and B. Nazer, "The approximate sum capacity of the symmetric Gaussian K-user interference channel," IEEE Trans. Inf. Theory, vol. 60, no. 6, pp. 3450-3482, Jun. 2014.
10. V. Ntranos, V. R. Cadambe, B. Nazer, and G. Caire, "Integer-forcing interference alignment," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Istanbul, Turkey, Jul. 2013, pp. 574-578.
11. M. Farag and B. Nazer, "The symmetric ergodic capacity of phase-fading interference channels to within a constant gap: 3 users in the strong and very strong regimes," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Istanbul, Turkey, Jul. 2013, pp. 764-768.
12. W. Nam, S.-Y. Chung, and Y. H. Lee, "Nested lattice codes for Gaussian relay networks with interference," IEEE Trans. Inf. Theory, vol. 57, no. 12, pp. 7733-7745, Dec. 2011.
13. A. S. Avestimehr, S. N. Diggavi, and D. N. C. Tse, "Wireless network information flow: A deterministic approach," IEEE Trans. Inf. Theory, vol. 57, no. 4, pp. 1872-1905, Apr. 2011.
14. C. Feng, D. Silva, and F. R. Kschischang, "An algebraic approach to physical-layer network coding," IEEE Trans. Inf. Theory, vol. 59, no. 11, pp. 7576-7596, Nov. 2013.
15. D. Micciancio and S. Goldwasser, Complexity of Lattice Problems: A Cryptographic Perspective (International International Series in Engineering and Computer Science), vol. 671. Cambridge, U.K.: Kluwer, 2002.
16. B. Nazer, "Successive compute-and-forward," in Proc. Int. Zurich Seminar Commun. (IZS), Zürich, Switzerland, Mar. 2012, pp. 1-4.
17. W. He, B. Nazer, and S. Shamai (Shitz), "Uplink-downlink duality for integer-forcing," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Honolulu, HI, USA, Jul. 2014, pp. 2544-2548.
18. S. H. Lim, C. Feng, A. Pastore, B. Nazer, and M. Gastpar. "A joint typicality approach to algebraic network information theory." IEEE Trans. Inf. Theory, [Online]. Available: http://arxiv.org/abs/1606.09548
19. R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, "Network information flow," IEEE Trans. Inf. Theory, vol. 46, no. 4, pp. 1204-1216, Jul. 2000.
20. S.-Y. R. Li, R. W. Yeung, and N. Cai, "Linear network coding," IEEE Trans. Inf. Theory, vol. 49, no. 2, pp. 371-381, Feb. 2003.
21. R. Koetter and M. Medard, "An algebraic approach to network coding," IEEE/ACM Trans. Netw., vol. 11, no. 5, pp. 782-795, Oct. 2003.
22. P. Popovski and H. Yomo, "The anti-packets can increase the achievable throughput of a wireless multi-hop network," in Proc. IEEE Int. Conf. Commun. (ICC), Istanbul, Turkey, Jun. 2006, pp. 3885-3890.
23. B. Nazer and M. Gastpar, "Computing over multiple-access channels with connections to wireless network coding," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Seattle, WA, USA, Jul. 2006, pp. 1354-1358.
24. S. Zhang, S. C. Liew, and P. Lam, "Hot topic: Physical-layer network coding," in Proc. 12th Annu. ACM Int. Conf. Mobile Comput. Netw. (MobiCom), Los Angeles, CA, USA, Sep. 2006, pp. 358-365.
25. B. Nazer and M. Gastpar, "Lattice coding increases multicast rates for Gaussian multiple-access networks," in Proc. 45th Annu. Allerton Conf. Commun., Control, Comput., Monticello, IL, USA, Sep. 2007, pp. 1-8.
26. M. P. Wilson, K. Narayanan, H. D. Pfister, and A. Sprintson, "Joint physical layer coding and network coding for bidirectional relaying," IEEE Trans. Inf. Theory, vol. 56, no. 11, pp. 5641-5654, Nov. 2010.
27. N. E. Tunali, Y.-C. Huang, J. J. Boutros, and K. R. Narayanan, "Lattices over Eisenstein integers for compute-and-forward," IEEE Trans. Inf. Theory, vol. 61, no. 10, pp. 5306-5321, Oct. 2015.
28. J.-C. Belfiore and C. Ling, "The flatness factor in lattice network coding: Design criterion and decoding algorithm," in Proc. Int. Zurich Seminar Commun. (IZS), Zürich, Switzerland, Mar. 2012, p. 111.
29. Z. Faraji-Dana and P. Mitran, "On non-binary constellations for channelcoded physical-layer network coding," IEEE Trans. Wireless Commun., vol. 12, no. 1, pp. 312-319, Jan. 2013.
30. N. E. Tunali, K. R. Narayanan, and H. D. Pfister, "Spatially-coupled low density lattices based on construction a with applications to computeand- forward," in Proc. IEEE Inf. Theory Workshop (ITW), Seville, Spain, Sep. 2013, pp. 1-5.
31. Y.-C. Huang, K. R. Narayanan, and N. E. Tunali, "Multistage computeand- forward with multilevel lattice codes based on product constructions," IEEE Trans. Inf. Theory, e-prints. [Online]. Available: http://arxiv.org/abs/1401.2228
32. J. Körner and K. Marton, "How to encode the modulo-two sum of binary sources (Corresp.)," IEEE Trans. Inf. Theory, vol. 25, no. 2, pp. 219-221, Mar. 1979.
33. T. Philosof and R. Zamir, "On the loss of single-letter characterization: The dirty multiple access channel," IEEE Trans. Inf. Theory, vol. 55, no. 6, pp. 2442-2454, Jun. 2009.
34. T. Philosof, R. Zamir, U. Erez, and A. J. Khisti, "Lattice strategies for the dirty multiple access channel," IEEE Trans. Inf. Theory, vol. 57, no. 8, pp. 5006-5035, Aug. 2011.
35. I.-H. Wang, "Approximate capacity of the dirty multiple-access channel with partial state information at the encoders," IEEE Trans. Inf. Theory, vol. 58, no. 5, pp. 2781-2787, May 2012.
36. D. Krithivasan and S. S. Pradhan, "Lattices for distributed source coding: Jointly Gaussian sources and reconstruction of a linear function," IEEE Trans. Inf. Theory, vol. 55, no. 12, pp. 5628-5651, Dec. 2009.
37. D. Krithivasan and S. S. Pradhan, "Distributed source coding using Abelian group codes: A new achievable rate-distortion region," IEEE Trans. Inf. Theory, vol. 57, no. 3, pp. 1495-1519, Mar. 2011.
38. A. B. Wagner, "On distributed compression of linear functions," IEEE Trans. Inf. Theory, vol. 57, no. 1, pp. 79-94, Jan. 2011.
39. M. A. Maddah-Ali and D. Tse, "Interference neutralization in distributed lossy source coding," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Austin, TX, USA, Jun. 2010, pp. 166-170.
40. Y. Yang and Z. Xiong, "Distributed compression of linear functions: Partial sum-rate tightness and gap to optimal sum-rate," IEEE Trans. Inf. Theory, vol. 60, no. 5, pp. 2835-2855, May 2014.
41. U. Niesen and P. Whiting, "The degrees of freedom of compute-andforward," IEEE Trans. Inf. Theory, vol. 58, no. 8, pp. 5214-5232, Aug. 2012.
42. Y. Song and N. Devroye, "Lattice codes for the Gaussian relay channel: Decode-and-forward and compress-and-forward," IEEE Trans. Inf. Theory, vol. 59, no. 8, pp. 4927-4948, Aug. 2013.
43. U. Niesen, B. Nazer, and P. Whiting, "Computation alignment: Capacity approximation without noise accumulation," IEEE Trans. Inf. Theory, vol. 59, no. 6, pp. 3811-3832, Jun. 2013.
44. Z. Ren, J. Goseling, J. H. Weber, and M. Gastpar, "Maximum throughput gain of compute-and-forward for multiple unicast," IEEE Commun. Lett., vol. 18, no. 7, pp. 1111-1114, Jul. 2014.
45. G. Bresler, A. Parekh, and D. N. C. Tse, "The approximate capacity of the many-to-one and one-to-many Gaussian interference channels," IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4566-4592, Sep. 2010.
46. A. S. Motahari, S. Oveis-Gharan, M.-A. Maddah-Ali, and A. K. Khandani, "Real interference alignment: Exploiting the potential of single antenna systems," IEEE Trans. Inf. Theory, vol. 60, no. 8, pp. 4799-4810, Aug. 2014.
47. U. Niesen and M. A. Maddah-Ali, "Interference alignment: From degrees of freedom to constant-gap capacity approximations," IEEE Trans. Inf. Theory, vol. 59, no. 8, pp. 4855-4888, Aug. 2013.
48. I. Shomorony and A. S. Avestimehr, "Degrees of freedom of two-hop wireless networks: Everyone gets the entire cake," IEEE Trans. Inf. Theory, vol. 60, no. 5, pp. 2417-2431, May 2014.
49. A. Padakandla, A. G. Sahebi, and S. S. Pradhan, "An achievable rate region for the three-user interference channel based on coset codes," IEEE Trans. Inf. Theory, vol. 62, no. 3, pp. 1250-1279, Mar. 2016.
50. X. He and A. Yener, "Providing secrecy with structured codes: Twouser Gaussian channels," IEEE Trans. Inf. Theory, vol. 60, no. 4, pp. 2121-2138, Apr. 2014.
51. G. Bagherikaram, A. S. Motahari, and A. K. Khandani, "On the secure DoF of the single-antenna MAC," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Austin, TX, USA, Jun. 2010, pp. 2588-2592.
52. S. Vatedka, N. Kashyap, and A. Thangaraj, "Secure compute-andforward in a bidirectional relay," IEEE Trans. Inf. Theory, vol. 61, no. 5, pp. 2531-2556, May 2015.
53. J. Xie and S. Ulukus, "Secure degrees of freedom of one-hop wireless networks," IEEE Trans. Inf. Theory, vol. 60, no. 6, pp. 3359-3378, Jun. 2014.
54. R. Zamir and M. Feder, "On lattice quantization noise," IEEE Trans. Inf. Theory, vol. 42, no. 4, pp. 1152-1159, Jul. 1996.
55. H.-A. Loeliger, "Averaging bounds for lattices and linear codes," IEEE Trans. Inf. Theory, vol. 43, no. 6, pp. 1767-1773, Nov. 1997.
56. G. D. Forney, Jr., M. D. Trott, and S.-Y. Chung, "Sphere-boundachieving coset codes and multilevel coset codes," IEEE Trans. Inf. Theory, vol. 46, no. 3, pp. 820-850, May 2000.
57. R. Zamir, S. Shamai (Shitz), and U. Erez, "Nested linear/lattice codes for structured multiterminal binning," IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 1250-1276, Jun. 2002.
58. U. Erez, S. Litsyn, and R. Zamir, "Lattices which are good for (almost) everything," IEEE Trans. Inf. Theory, vol. 51, no. 10, pp. 3401-3416, Oct. 2005.
59. O. Ordentlich and U. Erez, "A simple proof for the existence of 'good' pairs of nested lattices," IEEE Trans. Inf. Theory, vol. 62, no. 8, pp. 4439-4453, Aug. 2016.
60. R. Zamir, Lattice Coding for Signals and Networks. Cambridge, U.K.: Cambridge Univ. Press, 2014.
61. B. Nazer and R. Zamir, "Gaussian networks," in Lattice Coding for Signals and Networks: A Structured Coding Approach to Quantization, Modulation and Multiuser Information Theory. Cambridge, U.K.: Cambridge Univ. Press, 2014.
62. J. Zhu and M. Gastpar, "Multiple access via compute-andforward," IEEE Trans. Inf. Theory, e-prints. [Online]. Available: http://arxiv.org/abs/1407.8463
63. S.-N. Hong and G. Caire, "On interference networks over finite fields," IEEE Trans. Inf. Theory, vol. 60, no. 8, pp. 4902-4921, Aug. 2014.
64. S.-N. Hong and G. Caire, "Virtual full-duplex relaying with halfduplex relays," IEEE Trans. Inf. Theory, vol. 61, no. 9, pp. 4700-4720, Sep. 2015.
65. S.-N. Hong and G. Caire, "Structured lattice codes for 2'2'2 MIMO interference channel," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Istanbul, Turkey, Jul. 2013, pp. 2229-2233.
66. Y. Tan, X. Yuan, S. C. Liew, and A. Kavcic, "Asymmetric computeand- forward: Going beyond one hop," in Proc. 52nd Annu. Allerton Conf. Commun., Control Comput., Monticello, IL, USA, Sep./Oct. 2014, pp. 667-674.
67. B. Nazer, V. Cadambe, V. Ntranos, and G. Caire. (2015). "Expanding the compute-and-forward framework: Unequal powers, signal levels, and multiple linear combinations." [Online]. Available: http://arxiv.org/pdf/1504.01690v2.pdf
68. N. E. Tunali, K. R. Narayanan, J. J. Boutros, and Y.-C. Huang, "Lattices over Eisenstein integers for compute-and-forward," in Proc. 50th Annu. Allerton Conf. Commun., Control, Comput., Monticello, IL, USA, Oct. 2012, pp. 33-40.
69. D. J. Costello, Jr., and G. D. Forney, "Channel coding: The road to channel capacity," Proc. IEEE, vol. 95, no. 6, pp. 1150-1177, Jun. 2007.
70. A. Padakandla and S. S. Pradhan, "Achievable rate region based on coset codes for multiple access channels with states," IEEE Trans. Inf. Theory, e-prints. [Online]. Available: http://arxiv.org/abs/1301.5655
71. B. Nazer, A. Sanderovich, M. Gastpar, and S. Shamai (Shitz), "Structured superposition for backhaul constrained cellular uplink," in Proc. Int. Symp. Inf. Theory (ISIT), Seoul, South Korea, Jun./Jul. 2009, pp. 1530-1534.
72. D. A. Harville, Matrix Algebra From a Statistician's Perspective. New York, NY, USA: Springer-Verlag, 1997.
73. W. Nam, S.-Y. Chung, and Y. H. Lee, "Capacity of the Gaussian twoway relay channel to within 12 bit," IEEE Trans. Inf. Theory, vol. 56, no. 11, pp. 5488-5494, Nov. 2010.
74. D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge, U.K.: Cambridge Univ. Press, 2005.
75. M. Nokleby, B. Nazer, B. Aazhang, and N. Devroye, "Relays that cooperate to compute," in Proc. IEEE Int. Symp. Wireless Commun. Syst. (ISWCS), Paris, France, Sep. 2012, pp. 266-270.
76. M. K. Varanasi and T. Guess, "Optimum decision feedback multiuser equalization with successive decoding achieves the total capacity of the Gaussian multiple-access channel," in Proc. 31st Asilomar Conf. Signals, Syst. Comput., Pacific Grove, CA, USA, Nov. 1997, pp. 1405-1409.
77. M. R. Bremner, Lattice Basis Reduction. Boca Raton, FL, USA: CRC Press, 2012.
78. T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. Hoboken, NJ, USA: Wiley, 2006.
79. A. El Gamal and Y.-H. Kim, Network Information Theory. Cambridge, U.K.: Cambridge Univ. Press, 2011.
80. B. Rimoldi and R. Urbanke, "A rate-splitting approach to the Gaussian multiple-access channel," IEEE Trans. Inf. Theory, vol. 42, no. 2, pp. 364-375, Mar. 1996.
81. L. Wang, E. CSaCsǒglu, and Y.-H. Kim, "Sliding-window superposition coding for interference networks," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Honolulu, HI, USA, Jun./Jul. 2014, pp. 2749-2753.
82. T. Han and K. Kobayashi, "A new achievable rate region for the interference channel," IEEE Trans. Inf. Theory, vol. 27, no. 1, pp. 49-60, Jan. 1981.
83. H. Sato, "The capacity of the Gaussian interference channel under strong interference (Corresp.)," IEEE Trans. Inf. Theory, vol. 27, no. 6, pp. 786-788, Nov. 1981.
84. M. A. Maddah-Ali, A. S. Motahari, and A. K. Khandani, "Communication over MIMO X channels: Interference alignment, decomposition, and performance analysis," IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3457-3470, Aug. 2008.
85. V. R. Cadambe and S. A. Jafar, "Interference alignment and degrees of freedom of the K-user interference channel," IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3425-3441, Aug. 2008.
86. P. L. Chebyshev, "Mémoire sur les nombres premiers," J. Math. Pures Appl., vol. 1, pp. 366-390, 1852.
87. J. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups. New York, NY, USA: Springer-Verlag, 1998.
88. A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász, "Factoring polynomials with rational coefficients," Math. Ann., vol. 261, no. 4, pp. 515-534, Dec. 1982.
89. I. El Bakoury, W. He, and B. Nazer, "Integer-forcing interference alignment: Iterative optimization via aligned lattice reduction," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Hong Kong, Jun. 2015, pp. 581-585.
90. R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed. New York, NY, USA: Cambridge Univ. Press, 2013.
91. J. W. S. Cassels, An Introduction to Diophantine Approximations. Cambridge, U.K.: Cambridge Univ. Press, 1957.
92. J. J. Sylvester, "On the relation between the minor determinants of linearly equivalent quadratic functions," Philos. Mag., vol. 1, no. 4, pp. 295-305, 1851.