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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
Communication in a wireless medium
SourceReceiver
NoiseInterferenceSynchronizationChannel parameters
Communication over a wireless medium
SourceReceiver
NoiseInterferenceSynchronizationChannel parameters
Communication over a wireless medium
SourceReceiver
NoiseInterferenceSynchronizationChannel parameters
Communication over a wireless medium
SourceReceiver
NoiseInterferenceSynchronizationChannel parametersCut-set bounds tight?
Communication over a general network
S
A
D
B
C
T
h1
h2
h3
h6
h8
h7h4
h5
• The capacity region for networks with Gaussian channels is still an open problem
Communication over a general network
S
A
D
B
C
T
h1
h2
h3
h6
h8
h7h4
h5
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
Communication over a general network
S
A
D
B
C
T
h1
h2
h3
h6
h8
h7h4
h5
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
Communication over a point-to-point channel
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
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
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
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. 2004E. Arıkan, “Channel polarization: A method for constructing capacity achieving codes for symmetric binary-input memoryless channels,” IEEETrans. Inform. Theory, July 2009
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
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. 2004E. Arıkan, “Channel polarization: A method for constructing capacity achieving codes for symmetric binary-input memoryless channels,” IEEETrans. Inform. Theory, July 2009A. 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
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
Communication over a point-to-point channel
1X 2X 3X 1nX nX
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
iX is an integer
. . .= = = = =
5 42 3 19 90 1 0 0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1
. . .
. . .
. . .
. . .
. . .
= = = = =
5 42 3 19 9
and we take its binary representation
54321
6
122 ,6 Pm
Communication over a point-to-point channel
1X 2X 3X 1nX nX
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
iX is an integer
. . .
0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1
. . .
. . .
. . .
. . .
. . .
= = = = =
5 42 3 19 9
and we take its binary representation
iN0 1 0
1Y 2Y 3Y 1nY nY. . .
00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1
. . .
. . .
. . .
. . .
. . .
0 1 0
Bit flips
54321
654321
6
122 ,6 Pm
0
Communication over a point-to-point channel
1X 2X 3X 1nX nX
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
iX is an integer
. . .
0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1
. . .
. . .
. . .
. . .
. . .
= = = = =
5 42 3 19 9
and we take its binary representation
0 1 054321
6iN
1Y 2Y 3Y 1nY nY
00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1
. . .
. . .
. . .
. . .
. . .
0 1 0
Bit flips
54321
6
. . .
122 ,6 Pm
0
Communication over a point-to-point channel
1X 2X 3X 1nX nX
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
iX is an integer
. . .
0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1
. . .
. . .
. . .
. . .
. . .
= = = = =
5 42 3 19 9
and we take its binary representation
iN0 1 0
54321
61Y 2Y 3Y 1nY nY
1 00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1
. . .
. . .
. . .
. . .
. . .
0 1 0
Bit flips
54321
6
Dependent bit flips
. . .
122 ,6 Pm
Communication over a point-to-point channel
1X 2X 3X 1nX nX
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
iX is an integer
. . .
0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1
. . .
. . .
. . .
. . .
. . .
= = = = =
5 42 3 19 9
and we take its binary representation
iN0 1 0
54321
61Y 2Y 3Y 1nY nY
1 00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1
. . .
. . .
. . .
. . .
. . .
0 1 0
Bit flips
54321
6
Dependent bit flips
. . .
Less noisy bit levels
Very noisy bit levels
122 ,6 Pm
Communication over a point-to-point channel
1X 2X 3X 1nX nX
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
iX is an integer
. . .
0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1
. . .
. . .
. . .
. . .
. . .
= = = = =
5 42 3 19 9
and we take its binary representation
iN0 1 0
54321
61Y 2Y 3Y 1nY nY
1 00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1
. . .
. . .
. . .
. . .
. . .
0 1 0
Bit flips
54321
6
. . .
Less noisy bit levels
Very noisy bit levels
Code to correct adversarial errors
122 ,6 Pm
Communication over a point-to-point channel
1X 2X 3X 1nX nX
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
iX is an integer
. . .
0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1
. . .
. . .
. . .
. . .
. . .
= = = = =
5 42 3 19 9
and we take its binary representation
iN0 1 0
54321
61Y 2Y 3Y 1nY nY
1 00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1
. . .
. . .
. . .
. . .
. . .
0 1 0
Bit flips
54321
6
. . .
Less noisy bit levels
Very noisy bit levels
Code to correct adversarial errors
122 ,6 Pm
Communication over a point-to-point channel
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
iN
1X 2X 3X 1nX nX
iX is an integer
. . .
0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1
. . .
. . .
. . .
. . .
. . .
= = = = =
5 42 3 19 9
and we take its binary representation
0 1 054321
61Y 2Y 3Y 1nY nY
1 00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1
. . .
. . .
. . .
. . .
. . .
0 1 0
Bit flips
54321
6
. . .
Less noisy bit levels
Very noisy bit levels
pj ≤ 2.6 2-j
Rj = 1-h(2pj)
7.64
CRRm
jj
Due to adversarial errors
122 ,6 Pm
Communication over a point-to-point channel
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
1Y 2Y 3Y 1nY nY
1 00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1
. . .
. . .
. . .
. . .
. . .
0 1 0
Bit flips
54321
6
. . .
Less noisy bit levels
Very noisy bit levels
Code to correct adversarial errors pj ≤ 2.6 2-j
Rj = 1-h(2pj)
7.64
CRRm
jj
Due to adversarial errors
122 ,6 Pm
Complexity: )2( jnROExponential!!!
Communication over a point-to-point channel
);(max12
YXICEX
iii NXY
iX iY
iN
)1,0(~ NN i
P 1log2
1
PEX i 2, ,
1Y 2Y 3Y 1nY nY
1 00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1
. . .
. . .
. . .
. . .
. . .
0 1 0
Bit flips
54321
6
. . .
Less noisy bit levels
Very noisy bit levels
Code to correct adversarial errors pj ≤ 2.6 2-j
Rj = 1-h(2pj)
7.64
CRRm
jj
Due to adversarial errors
122 ,6 Pm
Complexity: )2( jnROExponential!!!
1Y nYlog
0 01 01 00 00 0
0 054321
6. . .. . .. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . . nkY log nkY log)1(
0 01 01 00 00 0
0 0. . .. . .. . .
. . .
. . .
. . .
symbol symbol
. . .
. . .
. . .
. . .
. . .
. . . ntY log ntY log)1(
0 01 01 00 00 0
0 0. . .. . .. . .
. . .
. . .
. . .
Redundancy
symbol
)2( lognR jO
)2( log2 n
n
e OP
Complexity per bit level:
)()log
( 2nOnn
nO
)(nO
Complexity:
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):3. For each inner code, guess each possible codeword and (low-weight)
error pattern due to bit flips at any node to decode – polynomial number.
4. Use outer RS code to correct any inner code errors
Challenges:5. Correlated bit-flips – distinguish between noise and carry bit-flips6. Mapping operations at nodes convert low-weight bit-flips to high-
weight errors – but entropy is all that matters.7. Concentration results on the expected number of correlated bit flips.
Overall code complexity O(n22|V|)