Salman Avestimehr Theodoros Dikaliotis Cornell University California Institute

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

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

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

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

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

Communication over a general network A h 1 h 3 h 4 B h 7 S T h 2 h 5 h 8 h 6 D C • The capacity region for networks with Gaussian channels is still an open problem

Communication over a general network A h 1 h 3 h 4 B h 7 S T h 2 h 5 h 8 h 6 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

Communication over a general network A h 1 h 3 h 4 B h 7 S T h 2 h 5 h 8 h 6 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

Communication over a point-to-point channel

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

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

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

Communication over a point-to-point channel is an integerand we take its binary representation . . . = 42 3 19 9 . . . . 190 = = 5 = 1 0 0 1 1 09 0 1 0 0 1 = 1 42 = = = 0 1 0 03 0 0 0 1 1 = = 6 50 5 0 4 0 3 1 2 0 1 1

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

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

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

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

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

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

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

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

Communication over a point-to-point channel Complexity per bit level: Complexity: Redundancy Code to correct. . adversarial errors . . . . symbol . . . 06 05 04 03 02 01 0 0 1. 0. . 1. . 0. . . 0 0 0 0 1 0. . . 1 1. . . 0 0. . . 1 Bit flipssymbol . . . 0 0 0 0 1 1 . . . pj ≤ 2. 602 -j. R. j. = 1 -h(2 p 0 j). . . 0 Due to adversarial 0 errors . . . 1. . . 0 Complexity: . . . 0 Exponential!!!. . . 0 Less noisy bit levels 6 0 5 0 4 1 3 1 Very noisy 2 bit levels 0 1 0 . symbol

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 highweight errors – but entropy is all that matters. 3. Concentration results on the expected number of correlated bit flips. Overall code complexity O(n 22|V|)

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