Tracey Ho Sidharth Jaggi Tsinghua University Hongyi Yao California Institute of Technology Theodoros Dikaliotis California Institute of Technology Chinese

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



SourceReceiver



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


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


• Quantize-map and forward achieves rates within a constant gap from the capacity


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


• 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, ,


•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, ,


•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, ,


•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, ,


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


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


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


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


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


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


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


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


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


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


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

. . .



Code to correct adversarial errors

122 ,6 Pm


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


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

. . .



Code to correct adversarial errors

122 ,6 Pm


);(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


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

. . .



pj ≤ 2.6 2-j

Rj = 1-h(2pj)

7.64

CRRm

jj

Due to adversarial errors

122 ,6 Pm


);(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

. . .



Code to correct adversarial errors pj ≤ 2.6 2-j

Rj = 1-h(2pj)

7.64

CRRm

jj


122 ,6 Pm

Complexity: )2( jnROExponential!!!


);(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

. . .



Code to correct adversarial errors pj ≤ 2.6 2-j

Rj = 1-h(2pj)

7.64

CRRm

jj


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:


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|)

Questions?

Documents

Tracey Ho Sidharth Jaggi Tsinghua University Hongyi Yao California Institute of Technology Theodoros Dikaliotis California Institute of Technology Chinese