1. Tracey Ho Sidharth Jaggi Tsinghua University Hongyi Yao California Institute of Technology Theodoros Dikaliotis California Institute of Technology Chinese University of Hong Kong Cornell University Salman Avestimehr

2. Communication in a wireless medium Source Receiver Noise Interference Synchronization Channel parameters

3. Communication over a wireless medium Source Receiver Noise Interference Synchronization Channel parameters

4. Communication over a wireless medium Source Receiver Noise Interference Synchronization Channel parameters

5. Communication over a wireless medium Source Receiver Noise Interference Synchronization Channel parameters Cut-set bounds tight?

6. Communication over a general network S A D B C T h1h1 h2h2 h3h3 h6h6 h8h8 h7h7 h4h4 h5h5 The capacity region for networks with Gaussian channels is still an open problem

7. Communication over a general network S A D B C T h1h1 h2h2 h3h3 h6h6 h8h8 h7h7 h4h4 h5h5 S. Avestimehr, S. Diggavi and D. Tse, “Wireless network information flow”, to appear in IEEE Transactions on Information Theory The capacity region for networks with Gaussian channels is still an open problem Quantize-map and forward achieves rates within a constant gap from the capacity

8. Communication over a general network S A D B C T h1h1 h2h2 h3h3 h6h6 h8h8 h7h7 h4h4 h5h5 S. Avestimehr, S. Diggavi and D. Tse, “Wireless network information flow”, to appear in IEEE Transactions on Information Theory The capacity region for networks with Gaussian channels is still an open problem Quantize-map and forward achieves rates within a constant gap from the capacity Our goal: polynomial-complexity codes that achieve within a constant gap from the capacity of the network

9. Communication over a point-to-point channel

10. Lattice codes Uri Erez, Ram Zamir, “Achieving (1/2)*log(1 + SNR) on the AWGN Channel With Lattice Encoding and Decoding,” IEEE Trans. On Inform. Theory, Oct. 2004

11. Communication over a point-to-point channel Lattice codes Polar codes Uri Erez, Ram Zamir, “Achieving (1/2)*log(1 + SNR) on the AWGN Channel With Lattice Encoding and Decoding,” IEEE Trans. On Inform. Theory, Oct. 2004 E. Arıkan, “Channel polarization: A method for constructing capacity achieving codes for symmetric binary-input memoryless channels,” IEEE Trans. Inform. Theory, July 2009

12. Communication over a point-to-point channel Lattice codes Polar codes Superposition codes Uri Erez, Ram Zamir, “Achieving (1/2)*log(1 + SNR) on the AWGN Channel With Lattice Encoding and Decoding,” IEEE Trans. On Inform. Theory, Oct. 2004 E. Arıkan, “Channel polarization: A method for constructing capacity achieving codes for symmetric binary-input memoryless channels,” IEEE Trans. Inform. Theory, July 2009 A. R. Barron, A. Joseph, “Least Squares Superposition Codes of Moderate Dictionary Size, Reliable at Rates up tp Capacity,” IEEE Trans. On Inform. Theory, June 2004

13. Communication over a point-to-point channel is an integer... ===== 5423199 01000 00010 01001 10000 01110 10111... ===== 5423199 and we take its binary representation 5 4 3 2 1 6

14. Communication over a point-to-point channel is an integer... 00 00010 01001 10000 01110 10111 ===== 5423199 and we take its binary representation 010... 0 00011 11001 11000 00001 00011 010 Bit flips 5 4 3 2 1 6 5 4 3 2 1 60

15. Communication over a point-to-point channel is an integer... 00 00010 01001 10000 01110 10111 ===== 5423199 and we take its binary representation 010 5 4 3 2 1 60 00011 11001 11000 00001 00011... 010 Bit flips 5 4 3 2 1 6... 0

16. Communication over a point-to-point channel is an integer... 00 00010 01001 10000 01110 10111 ===== 5423199 and we take its binary representation 010 5 4 3 2 1 610 00011 11001 11000 00001 00011... 010 Bit flips 5 4 3 2 1 6 Dependent bit flips...

17. Communication over a point-to-point channel is an integer... 00 00010 01001 10000 01110 10111 ===== 5423199 and we take its binary representation 010 5 4 3 2 1 610 00011 11001 11000 00001 00011... 010 Bit flips 5 4 3 2 1 6 Dependent bit flips... Less noisy bit levels Very noisy bit levels

18. Communication over a point-to-point channel is an integer... 00 00010 01001 10000 01110 10111 ===== 5423199 and we take its binary representation 010 5 4 3 2 1 610 00011 11001 11000 00001 00011... 010 Bit flips 5 4 3 2 1 6... Less noisy bit levels Very noisy bit levels Code to correct adversarial errors

19. Communication over a point-to-point channel is an integer... 00 00010 01001 10000 01110 10111 ===== 5423199 and we take its binary representation 010 5 4 3 2 1 610 00011 11001 11000 00001 00011... 010 Bit flips 5 4 3 2 1 6... Less noisy bit levels Very noisy bit levels Code to correct adversarial errors

20. Communication over a point-to-point channel is an integer... 00 00010 01001 10000 01110 10111 ===== 5423199 and we take its binary representation 010 5 4 3 2 1 6 10 00011 11001 11000 00001 00011... 010 Bit flips 5 4 3 2 1 6... Less noisy bit levels Very noisy bit levels p j ≤ 2.6 2 -j R j = 1-h(2p j ) Due to adversarial errors

21. Communication over a point-to-point channel 10 00011 11001 11000 00001 00011... 010 Bit flips 5 4 3 2 1 6... Less noisy bit levels Very noisy bit levels Code to correct adversarial errors p j ≤ 2.6 2 -j R j = 1-h(2p j ) Due to adversarial errors Complexity: Exponential!!!

22. Communication over a point-to-point channel 10 00011 11001 11000 00001 00011... 010 Bit flips 5 4 3 2 1 6... Less noisy bit levels Very noisy bit levels Code to correct adversarial errors p j ≤ 2.6 2 -j R j = 1-h(2p j ) Due to adversarial errors Complexity: Exponential!!! 00 10 10 00 00 00 5 4 3 2 1 6... 00 10 10 00 00 00 symbol... 00 10 10 00 00 00 Redundancy symbol Complexity per bit level: Complexity:

23. Communication over a general network S A D B C T h1h1 h2h2 h3h3 h6h6 h8h8 h7h7 h4h4 h5h5 S. Avestimehr, S. Diggavi and D. Tse, “Wireless network information flow”, to appear in IEEE Transactions on Information Theory For every node i in the network where

24. Communication over a general network Encoding Strategy: 1.RS Outer code (only at source) 2.ADT random inner code at source and interior nodes, length log n. Decoding strategy at receiver(s): 1.For each inner code, guess each possible codeword and (low-weight) error pattern due to bit flips at any node to decode – polynomial number. 2.Use outer RS code to correct any inner code errors Challenges: 1.Correlated bit-flips – distinguish between noise and carry bit-flips 2.Mapping operations at nodes convert low-weight bit-flips to high- weight errors – but entropy is all that matters. 3.Concentration results on the expected number of correlated bit flips. Overall code complexity O(n 2 2 |V| )

25. Questions?