Tracey Ho

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

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 A B h3 h1 h4 h7 S T h2 h5 h8 h6 D C • The capacity region for networks with Gaussian channels is still an open problem

7. Communication over a general network A B h3 h1 h4 h7 S T h2 h5 h8 h6 D C • 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 • S. Avestimehr, S. Diggavi and D. Tse, “Wireless network information flow”, to appear in IEEE Transactions on Information Theory

8. Communication over a general network A B h3 h1 h4 h7 S T h2 h5 h8 h6 D C • 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 • S. Avestimehr, S. Diggavi and D. Tse, “Wireless network information flow”, to appear in IEEE Transactions on Information Theory

9. Communication over a point-to-point channel

10. Communication over a point-to-point channel • 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 and we take its binary representation . . . = = = = = 6 0 1 0 0 0 5 42 3 19 9 . . . 5 0 0 0 1 0 . . . 4 0 1 0 0 1 . . . 3 1 0 0 0 0 . . . 2 0 1 1 1 0 . . . 1 1 0 1 1 1 = = = = = 5 42 3 19 9

14. Communication over a point-to-point channel is an integer and we take its binary representation . . . . . . 6 0 1 0 0 0 6 0 1 0 0 0 . . . . . . 5 0 0 0 1 0 5 0 0 0 1 1 . . . . . . 4 0 1 0 0 1 4 1 1 0 0 1 . . . . . . 3 1 0 0 0 0 3 1 1 0 0 0 . . . . . . 2 0 1 1 1 0 2 0 0 0 0 1 . . . . . . 1 1 0 1 1 1 1 0 0 0 1 1 = = = = = Bit flips 5 42 3 19 9

15. Communication over a point-to-point channel is an integer and we take its binary representation . . . . . . 6 0 1 0 0 0 6 0 1 0 0 0 . . . . . . 5 0 0 0 1 0 5 0 0 0 1 1 . . . . . . 4 0 1 0 0 1 4 1 1 0 0 1 . . . . . . 3 1 0 0 0 0 3 1 1 0 0 0 . . . . . . 2 0 1 1 1 0 2 0 0 0 0 1 . . . . . . 1 1 0 1 1 1 1 0 0 0 1 1 = = = = = Bit flips 5 42 3 19 9

16. Communication over a point-to-point channel is an integer and we take its binary representation Dependent bit flips . . . . . . 6 0 1 0 0 0 6 0 1 0 1 0 . . . . . . 5 0 0 0 1 0 5 0 0 0 1 1 . . . . . . 4 0 1 0 0 1 4 1 1 0 0 1 . . . . . . 3 1 0 0 0 0 3 1 1 0 0 0 . . . . . . 2 0 1 1 1 0 2 0 0 0 0 1 . . . . . . 1 1 0 1 1 1 1 0 0 0 1 1 = = = = = Bit flips 5 42 3 19 9

17. Communication over a point-to-point channel is an integer and we take its binary representation Dependent bit flips . . . . . . 6 0 1 0 0 0 6 0 1 0 1 0 . . . . . . 5 0 0 0 1 0 5 0 0 0 1 1 . . . . . . 4 0 1 0 0 1 4 1 1 0 0 1 . . . . . . 3 1 0 0 0 0 3 1 1 0 0 0 Less noisy bit levels Very noisy bit levels . . . . . . 2 0 1 1 1 0 2 0 0 0 0 1 . . . . . . 1 1 0 1 1 1 1 0 0 0 1 1 = = = = = Bit flips 5 42 3 19 9

18. Communication over a point-to-point channel is an integer and we take its binary representation . . . Code to correct adversarial errors . . . 6 0 1 0 0 0 6 0 1 0 1 0 . . . . . . 5 0 0 0 1 0 5 0 0 0 1 1 . . . . . . 4 0 1 0 0 1 4 1 1 0 0 1 . . . . . . 3 1 0 0 0 0 3 1 1 0 0 0 Less noisy bit levels Very noisy bit levels . . . . . . 2 0 1 1 1 0 2 0 0 0 0 1 . . . . . . 1 1 0 1 1 1 1 0 0 0 1 1 = = = = = Bit flips 5 42 3 19 9

19. Communication over a point-to-point channel is an integer and we take its binary representation . . . Code to correct adversarial errors . . . 6 0 1 0 0 0 6 0 1 0 1 0 . . . . . . 5 0 0 0 1 0 5 0 0 0 1 1 . . . . . . 4 0 1 0 0 1 4 1 1 0 0 1 . . . . . . 3 1 0 0 0 0 3 1 1 0 0 0 Less noisy bit levels Very noisy bit levels . . . . . . 2 0 1 1 1 0 2 0 0 0 0 1 . . . . . . 1 1 0 1 1 1 1 0 0 0 1 1 = = = = = Bit flips 5 42 3 19 9

20. Communication over a point-to-point channel is an integer and we take its binary representation . . . . . . 6 0 1 0 0 0 6 0 1 pj ≤ 2.6 2-j 0 1 0 Due to adversarial errors . . . . . . Rj = 1-h(2pj) 5 0 0 0 1 0 5 0 0 0 1 1 . . . . . . 4 0 1 0 0 1 4 1 1 0 0 1 . . . . . . 3 1 0 0 0 0 3 1 1 0 0 0 Less noisy bit levels Very noisy bit levels . . . . . . 2 0 1 1 1 0 2 0 0 0 0 1 . . . . . . 1 1 0 1 1 1 1 0 0 0 1 1 = = = = = Bit flips 5 42 3 19 9

21. Communication over a point-to-point channel Code to correct adversarial errors . . . 6 0 1 0 1 0 pj ≤ 2.6 2-j Due to adversarial errors . . . Rj = 1-h(2pj) 5 0 0 0 1 1 . . . 4 1 1 0 0 1 . . . 3 1 1 0 0 0 Complexity: Less noisy bit levels Very noisy bit levels . . . 2 0 0 0 0 1 Exponential!!! . . . 1 0 0 0 1 1 Bit flips

22. Communication over a point-to-point channel Complexity per bit level: Complexity: Redundancy Code to correct adversarial errors . . . . . . . . . . . . . . . . . . 6 0 0 6 0 1 0 0 1 0 0 pj ≤ 2.6 2-j 0 0 Due to adversarial errors . . . . . . . . . . . . . . . . . . Rj = 1-h(2pj) 5 0 0 5 0 0 0 0 1 0 1 0 0 . . . . . . . . . . . . . . . . . . 4 1 0 4 1 1 0 1 0 0 1 1 0 . . . . . . . . . . . . . . . . . . 3 1 0 3 1 1 0 1 0 0 0 1 0 Complexity: Less noisy bit levels Very noisy bit levels . . . . . . . . . . . . . . . . . . 2 0 0 2 0 0 0 0 0 0 1 0 0 Exponential!!! . . . . . . . . . . . . . . . . . . 1 0 0 1 0 0 0 0 1 0 1 0 0 Bit flips symbol symbol symbol

23. Communication over a general network Encoding Strategy: RS Outer code (only at source) ADT random inner code at source and interior nodes, length log n. Decoding strategy at receiver(s): For each inner code, guess each possible codeword and (low-weight) error pattern due to bit flips at any node to decode – polynomial number. Use outer RS code to correct any inner code errors Challenges: Correlated bit-flips – distinguish between noise and carry bit-flips Mapping operations at nodes convert low-weight bit-flips to high-weight errors – but entropy is all that matters. Concentration results on the expected number of correlated bit flips. Overall code complexity O(n22|V|)

24. Questions?