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Binary Error Correcting Network Codes. Qiwen Wang, Sidharth Jaggi , Shuo -Yen Robert Li Institute of Network Coding (INC) The Chinese University of Hong Kong October 19, 2011 IEEE Information Theory Workshop 2011, Paraty , Brazil. Outline. 1. Motivation. Model. 2. 3. Main Results. - PowerPoint PPT Presentation
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Qiwen Wang, Sidharth Jaggi, Shuo-Yen Robert Li
Institute of Network Coding (INC)The Chinese University of Hong Kong
October 19, 2011
IEEE Information Theory Workshop 2011, Paraty, Brazil
Binary Error Correcting Network Codes
The Chinese University of Hong Kong Institute of Network Coding 1
The Chinese University of Hong Kong Institute of Network Coding 2
Outline
4
Motivation1
2
3
5
Model
Main Results
Discussion
Conclusion
The Chinese University of Hong Kong Institute of Network Coding 3
Motivation:Challenges of NC over Noisy Networks
varying noise level1
(p-ε,p+ε)
The Chinese University of Hong Kong Institute of Network Coding 4
Motivation:Challenges of NC over Noisy Networks
1
2errors propagate through mix-and-forward
error
varying noise level
The Chinese University of Hong Kong Institute of Network Coding 5
Motivation:Challenges of NC over Noisy Networks
coding kernels unknown a priori
1
2
3
1 2
3 4
5 6 7
8 9
[f1,3,f1,5][f2,4,f2,7]
[f3,6,f4,6]
[f6,8,f6,9]
varying noise level
errors propagate through mix-and-forward
The Chinese University of Hong Kong Institute of Network Coding 6
Network Model & Code Model
Alice Bob
Mincut = C
… …
The Chinese University of Hong Kong Institute of Network Coding 7
Network Model & Code Model
Alice Bob
C
n
⁞⁞
……
Mincut = C
2mF
The Chinese University of Hong Kong Institute of Network Coding 8
Network Model & Code Model
Alice Bob
Cm x n
X
……
.
.
.
.
.
.
.
.
.
Cm x n
Y
……
.
.
.
.
.
.
.
.
.
⁞⁞
α
β
2mF
The Chinese University of Hong Kong Institute of Network Coding 9
Finite Field & Binary Field I
S1 S2 …… Sn 2mF
S11S12…S1m S21S22…S2m …… Sn1Sn2…Snm mn bits
S11
S12...S1m
S21
S22...S2m
……
Sn1
Sn2...Snm
mx n binary matrix
n symbols over
One Packet:
The Chinese University of Hong Kong Institute of Network Coding 10
Finite Field & Binary Field II
T
2mFa symbol over
T11 T12 …… T1m
T21 T22 …… T2m
. . .Tm1 Tm2 …… Tmm
m x m binary matrix
TS
2mFMultiplication over
T11 T12 …… T1m
T21 T22 …… T2m
. . .Tm1 Tm2 …… Tmm
S11
S21
. . .Sm1
Multiplication over binary field
The Chinese University of Hong Kong Institute of Network Coding 11
Transfer Matrix
Noiseless NetworkX Y
YCm x nCm x Cm
T XCm x n
×
… …
The Chinese University of Hong Kong Institute of Network Coding 12
Noise Model:Worst-case Bit-flip Error
111 101 001 …… 110 101 011 ……
Link A
000 110 100 …… 000 101 100 ……
Link B
111 101 001 …… 100 100 011 ……
Link A
000 110 100 …… 000 110 100 ……
Link B
OR
Errors can be arbitrarily distributed, with an upper bound of fraction p.Worst possible damage can happen to received packets.
The Chinese University of Hong Kong Institute of Network Coding 13
Noise Model:Worst-case Bit-flip Error
Z……
.
.
.
.
.
.
.
.
.
Em x n
Worst-case bit-flip error matrix Z: no more than pEmn 1s, arbitrarily distributedE: num of edges in the network
The Chinese University of Hong Kong Institute of Network Coding 14
Noise Model:Worst-case Bit-flip Error
Z……
.
.
.
.
.
.
.
.
.
Em x n
Error bits on the 1st edge
111 101 001 …… 100 111 001 ……
Edge 1Worst-case bit-flip error matrix Z: no more than pEmn 1s, arbitrarily distributedE: num of edges in the network
The Chinese University of Hong Kong Institute of Network Coding 15
Noise Model:Worst-case Bit-flip Error
011
010
Z……
.
.
.
.
.
.
.
.
.
Em x n
Error bits on the 1st edge
111 101 001 …… 100 111 001 ……
Edge 1Worst-case bit-flip error matrix Z: no more than pEmn 1s, arbitrarily distributedE: num of edges in the network
The Chinese University of Hong Kong Institute of Network Coding 16
Impulse Response Matrix
Z
…
X Y
Y
Cm x nCm x Cm Cm x Em
T X ZT̂× ×
Em x n
Cm x n
… …
The Chinese University of Hong Kong Institute of Network Coding 17
Transform Metric
YT X ZT̂× ×
YTX ZT̂ ×
00101...0010
The Chinese University of Hong Kong Institute of Network Coding 18
Transform Metric
T̂
TX YXi Yi
+di columns
=
…
ˆ1
( , )n
iTi
d TX Y d
• di is the minimum number of columns of that need to be added to TX(i) to obtain Y(i).
• Claim: is a distance metric.
T̂
ˆ ( , )T
d TX Y
The Chinese University of Hong Kong Institute of Network Coding 19
Hamming-type Upper Bound
For all p less than C/(2Em), an upper bound on the achievable rate of any code over the worst-case binary-error network is
Theorem 1
1 ( )E H pC
The Chinese University of Hong Kong Institute of Network Coding 20
Hamming-type Upper Bound
• Total number of Cm x n binary matrices (volume of the big square) is .
• Lower bound of the volume of the balls
• Consider those Z’s where every column has pEm ones in it, distinct Z results in distinct .
• The number of distinct is at least ~
• Upper bound on the size of any codebook is
• Asymptotically in n, the Hamming-type upper bound is .
Proof (sketch)
pEmnpEmn
pEmn
Hamming 1 ( )ER H pC
ˆ ˆ( , ) { | ( , ) }T T
B TX pEmn Y d TX Y pEmn
T̂ZnEm
pEm
( ) log( 1)2H p Emn Em
T̂Z
2Cmn
log( 1)(1 ( ) )
( ) log( 1)
2 22
E EmCmn H p CmnC Cmn
EmnH p Em
1 ( ) EH pC
The Chinese University of Hong Kong Institute of Network Coding 21
Coherent/Non-coherent NC
Coherent NC: receiver knows the internal coding coefficients, hence knows T and .
Non-coherent NC: coding coefficients, hence T and , are unknown in advance, more realistic setting.
However, the random linear coding coefficients are usually chosen on the fly.
T̂
T̂
The Chinese University of Hong Kong Institute of Network Coding 22
Coherent/Non-coherent NC
Coherent NC: receiver knows the internal coding coefficients, hence knows T and .
Non-coherent NC: coding coefficients, hence T and , are unknown in advance, more realistic setting.
However, the random linear coding coefficients are usually chosen on the fly.
T̂
T̂
The Chinese University of Hong Kong Institute of Network Coding 23
Coherent/Non-coherent NC
Coherent NC: receiver knows the internal coding coefficients, hence knows T and .
Non-coherent NC: coding coefficients, hence T and , are unknown in advance, more realistic setting.
However, the random linear coding coefficients are usually chosen on the fly.
T̂
T̂
The Chinese University of Hong Kong Institute of Network Coding 24
Coherent/Non-coherent NC
Coherent NC: receiver knows the internal coding coefficients, hence knows T and .
Non-coherent NC: coding coefficients, hence T and , are unknown in advance, more realistic setting.
However, the random linear coding coefficients are usually chosen on the fly.
T̂
T̂
The Chinese University of Hong Kong Institute of Network Coding 25
GV-type Lower Bound
Coherent GV-type network codes achieve a rate of at least
Theorem 2
Non-coherent GV-type network codes achieve a rate of at least
Theorem 3
1 (2 )E H pC
1 (2 )E H pC
The Chinese University of Hong Kong Institute of Network Coding 26
GV-type Lower Bound
• Need an upper bound on volume of instead of the lower bound on volume of as in Thm1. (sphere packing vs. covering)
• Different Y, or equivalently , can be bounded above by the number of different Z, which equals
• The summation can be bounded from above by ~
• Lower bound on the size of the codebook
• Asymptotically in n, the rate of coherent GV-type NC .
Proof of Thm2 (sketch)
T̂Z
TX(1)
2pEmn
TX(2) 2pEmn
TX(3)
2pEmn
GV-coherent 1 (2 )ER H pC
ˆ ( , 2 )T
B TX pEmnˆ ( , )
TB TX pEmn
2
0
pEmn
i
Emni
(2 1)2
EmnpEmn
pEmn
(2 )(2 1)2H p EmnpEmn
log(2 1)(1 (2 ) )
(2 )
2 2(2 1)2
E pEmnCmn H p CmnC n
H p EmnpEmn
1 (2 ) EH pC
The Chinese University of Hong Kong Institute of Network Coding 27
GV-type Lower Bound
• Crucial difference with the proof of Thm2: the process of choosing codewords.
• Consider all possible values of , at most (and hence T, since it comprises of a specific subset of C columns of ).
• The number of potential codewords that can be chosen in the codebook is at least
which equals
• Asymptotically in n, it leads to the same rate of as coherent NC in Theorem2.
Proof of Thm3 (sketch)
T̂T̂
2CEm
(2 )
22 (2 1)2
Cmn
CEm H p EmnpEmn
log(2 1)(1 (2 ) )2
E pEmn EH p CmnC n
1 (2 )E H pC
The Chinese University of Hong Kong Institute of Network Coding 28
Scale of Parameters
For all p less than ,the Hamming-type and GV-type bounds hold.
Claim
• Theorem 1 (Hamming-type upper bound)requires that .• For the GV-type bound in Thm2 and Thm3 to give non-negative rates, .
When p is small,
Proof
1
1min( , )2 2m
CEm
2CpEm
(2 ) 1EH pC
1(2 ) 2 (log )2
1(log )2
log 2
1
E EH p pC p C
C EEm p C
pm
The Chinese University of Hong Kong Institute of Network Coding 29
Conclusion
Coherent/non-coherent GV-type lower bound:
GV-type codes: End-to-end natureComplexity: poly. in block lengthHamming-type upper
bound:
Worst-case bit-flip error model
1 ( ) EH pC
1 (2 ) EH pC
The Chinese University of Hong Kong Institute of Network Coding 30
Future Direction
• Efficient coding schemes• Other binary noise model• Combine link-by-link codeswith our end-to-end codes
The Chinese University of Hong Kong Institute of Network Coding 31
Thank you!
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