new communication strategies for broadcast and interference … · broadcast communication nested...
TRANSCRIPT
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
New communication strategies for
broadcast and interference networks
S. Sandeep Pradhan
(Joint work with Arun Padakandla and Aria Sahebi)
University of Michigan, Ann Arbor
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Distributed Information Coding
Proliferation of wireless data and sensor network applications
Supported by distributed information processing
Information-theoretic perspective
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
1: Distributed Field Gathering
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
2: Broadcast and Interference Networks
Mobile Transmitter
1
Mobile Transmitter
2
Mobile
2
Mobile
1
Receiver
Receiver
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Information and Coding theory: Tradition
Information Theory:
Develop efficient communication strategies
No constraints on memory/computation for
encoding/decoding
Obtain performance limits that are independent of technology
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Information and Coding theory: Tradition
Information Theory:
Develop efficient communication strategies
No constraints on memory/computation for
encoding/decoding
Obtain performance limits that are independent of technology
Coding Theory:
Approach these limits using algebraic codes (Ex: linear codes)
Fast encoding and decoding algorithms
Objective: practical implementability of optimal
communication systems
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Information theory: Orders of magnitude
Subatomic scale: 10−23 − 10−15 Physicists
Atomic scale: 10−15 − 10−6 Chemists
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Information theory: Orders of magnitude
Subatomic scale: 10−23 − 10−15 Physicists
Atomic scale: 10−15 − 10−6 Chemists
Human scale: 10−6 − 106 Biologists
Astronomical scale: 106 − 1027 Astromoners
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Information theory: Orders of magnitude
Subatomic scale: 10−23 − 10−15 Physicists
Atomic scale: 10−15 − 10−6 Chemists
Human scale: 10−6 − 106 Biologists
Astronomical scale: 106 − 1027 Astromoners
Information-theory scale: 10n, n sufficiently large.
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Probability versus Algebra
Information Theory Tools: based on probability
Finding the optimal communication system directly is difficult
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Probability versus Algebra
Information Theory Tools: based on probability
Finding the optimal communication system directly is difficult
Random Coding:
Build a collection of communication systems (ensemble)
Put a probability distribution on them
Show good average performance
Craft ensembles using probability
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Probability versus Algebra
Information Theory Tools: based on probability
Finding the optimal communication system directly is difficult
Random Coding:
Build a collection of communication systems (ensemble)
Put a probability distribution on them
Show good average performance
Craft ensembles using probability
Coding Theory Tools: Abstract algebra (groups, fields)
Exploit algebraic structure to develop algorithms of
polynomial complexity for encoding/decoding
Study a very small ensemble at a time.
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Random Coding in networks
Prob. distribution on a collection of codebooks (ensemble)
Extensions of Shannon ensembles
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Random Coding in networks
Prob. distribution on a collection of codebooks (ensemble)
Extensions of Shannon ensembles
Lot of bad codebooks in the ensemble
Average performance significantly affected by these bad codes
Do not achieve optimality in general
Many problems have remained open for decades.
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Coding theory to the rescue ?
It turns out that algebraic structure can be used to weed out
bad codes
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Coding theory to the rescue ?
It turns out that algebraic structure can be used to weed out
bad codes
Gain barely noticeable in point-to-point communication
Improvement in second order performance (error exponents)
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Coding theory to the rescue ?
It turns out that algebraic structure can be used to weed out
bad codes
Gain barely noticeable in point-to-point communication
Improvement in second order performance (error exponents)
Gains significant in multi-terminal communication
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Coding theory to the rescue ?
It turns out that algebraic structure can be used to weed out
bad codes
Gain barely noticeable in point-to-point communication
Improvement in second order performance (error exponents)
Gains significant in multi-terminal communication
Time for Question?
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Broadcast Networks
Mobile
2
Mobile
1
Receiver
Receiver
Station
Base
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Point-to-point communication
Start with Binary symmetric channel
δ
δ
0 0
11
1−δ
1−δ
+YX
N
N ∼ Be(δ), and + is addition modulo 2
Capacity = maxP(x) I (X ;Y ) = 1− h(δ).
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of an optimal code
Output is within a ball around a transmitted codeword
Maximum likelyhood decoding
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of an optimal code
Output is within a ball around a transmitted codeword
Maximum likelyhood decoding
Time for Question?
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Twitter and Eddington Number
Suppose you to want tweet on a BSC:
140 characters
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Twitter and Eddington Number
Suppose you to want tweet on a BSC:
140 characters
Entropy of tweets = 1.9 bits/character, ⇒ 266 bits.
Suppose δ = 0.11, then C = 0.5 bits/channel use
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Twitter and Eddington Number
Suppose you to want tweet on a BSC:
140 characters
Entropy of tweets = 1.9 bits/character, ⇒ 266 bits.
Suppose δ = 0.11, then C = 0.5 bits/channel use
A tweet can be sent by using BSC 532 times.
Number of possible tweets = 2266
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Twitter and Eddington Number
Suppose you to want tweet on a BSC:
140 characters
Entropy of tweets = 1.9 bits/character, ⇒ 266 bits.
Suppose δ = 0.11, then C = 0.5 bits/channel use
A tweet can be sent by using BSC 532 times.
Number of possible tweets = 2266
Equals the number of protons in the observable universe
Named after Arthur Eddington.
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constaint
1nEwH(X
n) ≤ q
i.e., a codeword has at most q fractions of 1’s
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constaint
1nEwH(X
n) ≤ q
i.e., a codeword has at most q fractions of 1’s
Capacity-cost function
C (q) = maxEwH(X )≤q
I (X ;Y ) = H(Y )−H(Y |X ) = h(q ∗δ)−h(δ)
q ∗ δ = (1− q)δ + q(1− δ)
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constaint
1nEwH(X
n) ≤ q
i.e., a codeword has at most q fractions of 1’s
Capacity-cost function
C (q) = maxEwH(X )≤q
I (X ;Y ) = H(Y )−H(Y |X ) = h(q ∗δ)−h(δ)
q ∗ δ = (1− q)δ + q(1− δ)
X ∼ Be(q)
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of an optimal code
Big circle: the set of all words with q fraction of 1’s
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constraint and interference
+X
S
+
N
Y
S ∼ Be(0.5) and N ∼ Be(δ)
S is non-causally observable only at encoder
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constraint and interference
+X
S
+
N
Y
S ∼ Be(0.5) and N ∼ Be(δ)
S is non-causally observable only at encoder
1nEwH(X
n) ≤ q
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Applications
Digital watermarking, data hiding, covert communication
+X
S
+
N
YEncoder Decoder
Original Image
❁❇❏❑❉❒❈❇❑❏❇❁❇❏❑❉❒❈❇❑❏❇
Big Govt.
(on youtube)Watermark
Blind watermarking
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Applications
Digital watermarking, data hiding, covert communication
+X
S
+
N
YEncoder Decoder
Original Image
❁❇❏❑❉❒❈❇❑❏❇❁❇❏❑❉❒❈❇❑❏❇
Big Govt.
(on youtube)Watermark
Blind watermarking
You want big govt. but you dont trust it too much
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constraint and interference
Q1: What is the communication strategy?
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constraint and interference
Q1: What is the communication strategy?
A1. Try cancelling it
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constraint and interference
Q1: What is the communication strategy?
A1. Try cancelling it
You cannot, you do not have enough number of ones.
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constraint and interference
Q1: What is the communication strategy?
A1. Try cancelling it
You cannot, you do not have enough number of ones.
A2. Ride on the interference
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constraint and interference
Q1: What is the communication strategy?
A1. Try cancelling it
You cannot, you do not have enough number of ones.
A2. Ride on the interference
Nudge the interference with channel input toward a codeword
But, you have got just q fraction of ones.
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constraint and interference
Q1: What is the communication strategy?
A1. Try cancelling it
You cannot, you do not have enough number of ones.
A2. Ride on the interference
Nudge the interference with channel input toward a codeword
But, you have got just q fraction of ones.
Gelfand-Pinsker: Nudge toward a codeword from a set
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
BSC with cost constraint and interference
Q1: What is the communication strategy?
A1. Try cancelling it
You cannot, you do not have enough number of ones.
A2. Ride on the interference
Nudge the interference with channel input toward a codeword
But, you have got just q fraction of ones.
Gelfand-Pinsker: Nudge toward a codeword from a set
Q2. How large should the set be?
Rate of the set: 1− h(q).
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of an optimal set of codewords
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of an optimal set of codewords
All these codewords are assigned for a message
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of an optimal set of codewords
All these codewords are assigned for a message
Select a codeword to which you can nudge the interference..
..by spending just q fraction of ones ⇒ U = X + S
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of an optimal set of codewords
All these codewords are assigned for a message
Select a codeword to which you can nudge the interference..
..by spending just q fraction of ones ⇒ U = X + S
New effective channel: Y = U + N with capacity 1− h(δ)
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Precoding for Interference
Rate of the composite codebook: 1− h(δ)
Rate of a sub-code-book: 1− h(q)
Transmission rate: difference = h(q)− h(δ)
Capacity in general case [Gelfand-Pinsker ’80]
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Precoding for Interference
Rate of the composite codebook: 1− h(δ)
Rate of a sub-code-book: 1− h(q)
Transmission rate: difference = h(q)− h(δ)
Capacity in general case [Gelfand-Pinsker ’80]
C (q) = maxP(U,X |S):EwH(X )≤q
I (U;Y )− I (U; S)
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of Capacity cost function
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
0.6
|
|
|
|
|
|
|
|
|
|q’
Cost q
Rat
e h(q* )−h( )
k(q)=h(q)−h( )C(q)
δ δ
δ
Bottomline: Rate loss as compared to no inferference
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Broadcast Channel: Cover ’72
Broadcast
Channel
M M1 2
Encoder
Decoder 1
Decoder 2
M
M
1
2
X
Y
Z
Channel with one input and multiple outputs
Same signal should contain info. meant for both receivers
Capacity region still not known in general
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Broadcast Channel: Cover ’72
Broadcast
Channel
M M1 2
Encoder
Decoder 1
Decoder 2
M
M
1
2
X
Y
Z
Channel with one input and multiple outputs
Same signal should contain info. meant for both receivers
Capacity region still not known in general
Time for questions?
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Marton’s Coding Strategy: Two receivers
Create a signal that carry information for the second receiver
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Marton’s Coding Strategy: Two receivers
Create a signal that carry information for the second receiver
This signal acts as interference for the signal of the first
How to tackle (self) interference?
Broadcast and interference
![Page 53: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/53.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Marton’s Coding Strategy: Two receivers
Create a signal that carry information for the second receiver
This signal acts as interference for the signal of the first
How to tackle (self) interference?
Make the first receiver decode a large portion of interference
This portion is given by a (univariate) function
Broadcast and interference
![Page 54: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/54.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Marton’s Coding Strategy: Two receivers
Create a signal that carry information for the second receiver
This signal acts as interference for the signal of the first
How to tackle (self) interference?
Make the first receiver decode a large portion of interference
This portion is given by a (univariate) function
The rest is precoded for using Gelfand-Pinsker strategy
Broadcast and interference
![Page 55: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/55.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Marton’s Coding Strategy: Two receivers
Create a signal that carry information for the second receiver
This signal acts as interference for the signal of the first
How to tackle (self) interference?
Make the first receiver decode a large portion of interference
This portion is given by a (univariate) function
The rest is precoded for using Gelfand-Pinsker strategy
This strategy is optimal for many special cases
We do not know whether it is optimal in general
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Example: so-called non-degraded channel
+
+
Y
Z
N
N
X
X
1
1
2
2
N1 ∼ Be(δ), and N2 ∼ Be(ǫ), and no constraint on X2
Hamming weight constraint on X1:1nEwH(X
n1 ) ≤ q
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Example: so-called non-degraded channel
+
+
Y
Z
N
N
X
X
1
1
2
2
N1 ∼ Be(δ), and N2 ∼ Be(ǫ), and no constraint on X2
Hamming weight constraint on X1:1nEwH(X
n1 ) ≤ q
Fix R2 = 1− h(ǫ), and assume δ < ǫ
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Example: so-called non-degraded channel
+
+
Y
Z
N
N
X
X
1
1
2
2
N1 ∼ Be(δ), and N2 ∼ Be(ǫ), and no constraint on X2
Hamming weight constraint on X1:1nEwH(X
n1 ) ≤ q
Fix R2 = 1− h(ǫ), and assume δ < ǫ
When q ∗ δ ≤ ǫ, Rec. 1 can decode interference completely
Broadcast and interference
![Page 59: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/59.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Example: so-called non-degraded channel
+
+
Y
Z
N
N
X
X
1
1
2
2
N1 ∼ Be(δ), and N2 ∼ Be(ǫ), and no constraint on X2
Hamming weight constraint on X1:1nEwH(X
n1 ) ≤ q
Fix R2 = 1− h(ǫ), and assume δ < ǫ
When q ∗ δ ≤ ǫ, Rec. 1 can decode interference completely
a.k.a no interference ⇒ R1 = h(q ∗ δ)− h(δ)
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Example: so-called non-degraded channel
+
+
Y
Z
N
N
X
X
1
1
2
2
N1 ∼ Be(δ), and N2 ∼ Be(ǫ), and no constraint on X2
Hamming weight constraint on X1:1nEwH(X
n1 ) ≤ q
Fix R2 = 1− h(ǫ), and assume δ < ǫ
When q ∗ δ ≤ ǫ, Rec. 1 can decode interference completely
a.k.a no interference ⇒ R1 = h(q ∗ δ)− h(δ)
Otherwise, precode for X2: ⇒ R1 = h(q)− h(δ)
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of Rate Region
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Cost q
Rat
e
Decode a univariate function of interference & precode for the rest
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Broadcast with more receivers
Marton’s strategy can be easily extended
Consider 3 receiver case: At receiver 1:
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Broadcast with more receivers
Marton’s strategy can be easily extended
Consider 3 receiver case: At receiver 1:
(self) interference of signals of Rec. 2 and Rec. 3
Broadcast and interference
![Page 64: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/64.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Broadcast with more receivers
Marton’s strategy can be easily extended
Consider 3 receiver case: At receiver 1:
(self) interference of signals of Rec. 2 and Rec. 3
Decode a univariate function of signal meant for Rec. 2...
.. and a univariate function of signal meant for Rec. 3.
Broadcast and interference
![Page 65: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/65.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Broadcast with more receivers
Marton’s strategy can be easily extended
Consider 3 receiver case: At receiver 1:
(self) interference of signals of Rec. 2 and Rec. 3
Decode a univariate function of signal meant for Rec. 2...
.. and a univariate function of signal meant for Rec. 3.
Precode for the rest
Broadcast and interference
![Page 66: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/66.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Broadcast with more receivers
Marton’s strategy can be easily extended
Consider 3 receiver case: At receiver 1:
(self) interference of signals of Rec. 2 and Rec. 3
Decode a univariate function of signal meant for Rec. 2...
.. and a univariate function of signal meant for Rec. 3.
Precode for the rest
All these being done using random codes
No need for linear or algebraic codes till now
Broadcast and interference
![Page 67: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/67.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Broadcast with more receivers
Marton’s strategy can be easily extended
Consider 3 receiver case: At receiver 1:
(self) interference of signals of Rec. 2 and Rec. 3
Decode a univariate function of signal meant for Rec. 2...
.. and a univariate function of signal meant for Rec. 3.
Precode for the rest
All these being done using random codes
No need for linear or algebraic codes till now
We can show that such a strategy is strictly suboptimal
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
New Strategy
Decode a bivariate function of the signals meant for other two
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
New Strategy
Decode a bivariate function of the signals meant for other two
It turns out that to exploit this we need linear codes
Broadcast and interference
![Page 70: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/70.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
New Strategy
Decode a bivariate function of the signals meant for other two
It turns out that to exploit this we need linear codes
+
+
+X
X
X
N
N
N
2
3
1
2
1
3 A
Z
Y
N2,N3 ∼ Be(ǫ), and no constraints on X2 and X3
N1 ∼ Be(δ) and the usual : 1nEwH(X
n1 ) ≤ q
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
New Strategy
Decode a bivariate function of the signals meant for other two
It turns out that to exploit this we need linear codes
+
+
+X
X
X
N
N
N
2
3
1
2
1
3 A
Z
Y
N2,N3 ∼ Be(ǫ), and no constraints on X2 and X3
N1 ∼ Be(δ) and the usual : 1nEwH(X
n1 ) ≤ q
Let R2 = R3 = 1− h(ǫ), the incorrigble brutes!
Let δ < ǫ
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Deficiency of random codes
δ = 0.1 and ǫ = 0.2
+δ ε
0.5310+
0.5562
ε+
0.2781
1−h( ) 1−h( ) 2(1−h( ))
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Deficiency of random codes
δ = 0.1 and ǫ = 0.2
+δ ε
0.5310+
0.5562
ε+
0.2781
1−h( ) 1−h( ) 2(1−h( ))
Marton wishes to decode “full” interference: (X2,X3):
1− h(q ∗ δ) > 2(1− h(ǫ))
Broadcast and interference
![Page 74: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/74.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Deficiency of random codes
δ = 0.1 and ǫ = 0.2
+δ ε
0.5310+
0.5562
ε+
0.2781
1−h( ) 1−h( ) 2(1−h( ))
Marton wishes to decode “full” interference: (X2,X3):
1− h(q ∗ δ) > 2(1− h(ǫ))
a.k.a never going to happen
Marton ends up doing precoding incurring rate loss
Broadcast and interference
![Page 75: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/75.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Deficiency of random codes
δ = 0.1 and ǫ = 0.2
+δ ε
0.5310+
0.5562
ε+
0.2781
1−h( ) 1−h( ) 2(1−h( ))
Marton wishes to decode “full” interference: (X2,X3):
1− h(q ∗ δ) > 2(1− h(ǫ))
a.k.a never going to happen
Marton ends up doing precoding incurring rate loss
New Approach: Try decoding actual interference: X2 + X3
Broadcast and interference
![Page 76: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/76.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Deficiency of random codes
δ = 0.1 and ǫ = 0.2
+δ ε
0.5310+
0.5562
ε+
0.2781
1−h( ) 1−h( ) 2(1−h( ))
Marton wishes to decode “full” interference: (X2,X3):
1− h(q ∗ δ) > 2(1− h(ǫ))
a.k.a never going to happen
Marton ends up doing precoding incurring rate loss
New Approach: Try decoding actual interference: X2 + X3
Benefit if the range of X2 + X3 is ≪ range of (X2,X3)
Broadcast and interference
![Page 77: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/77.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Deficiency of random codes
δ = 0.1 and ǫ = 0.2
+δ ε
0.5310+
0.5562
ε+
0.2781
1−h( ) 1−h( ) 2(1−h( ))
Marton wishes to decode “full” interference: (X2,X3):
1− h(q ∗ δ) > 2(1− h(ǫ))
a.k.a never going to happen
Marton ends up doing precoding incurring rate loss
New Approach: Try decoding actual interference: X2 + X3
Benefit if the range of X2 + X3 is ≪ range of (X2,X3)
If X2 and X3 are “random”, this wont happen
Broadcast and interference
![Page 78: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/78.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of sum of two random sets
+
=
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of sum of two cosets of a linear code
+
=
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Exploits of Linear Codes
The “incorrigible brutes” can have their capacities
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Exploits of Linear Codes
The “incorrigible brutes” can have their capacities
We just need their codebooks to behave “algebraic”
We know that linear codes achieve the capacity of BSC
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Exploits of Linear Codes
The “incorrigible brutes” can have their capacities
We just need their codebooks to behave “algebraic”
We know that linear codes achieve the capacity of BSC
rate of X2 = rate of X3 = rate of X2 + X3 = 1− h(ǫ)
Since δ < ǫ, we have for small q: q ∗ δ < ǫ
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Exploits of Linear Codes
The “incorrigible brutes” can have their capacities
We just need their codebooks to behave “algebraic”
We know that linear codes achieve the capacity of BSC
rate of X2 = rate of X3 = rate of X2 + X3 = 1− h(ǫ)
Since δ < ǫ, we have for small q: q ∗ δ < ǫ
Hence 1− h(q ∗ δ) > 1− h(ǫ)
Rec. 1 can decode the actual interference and subtract it off
Broadcast and interference
![Page 84: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/84.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Exploits of Linear Codes
The “incorrigible brutes” can have their capacities
We just need their codebooks to behave “algebraic”
We know that linear codes achieve the capacity of BSC
rate of X2 = rate of X3 = rate of X2 + X3 = 1− h(ǫ)
Since δ < ǫ, we have for small q: q ∗ δ < ǫ
Hence 1− h(q ∗ δ) > 1− h(ǫ)
Rec. 1 can decode the actual interference and subtract it off
Then decodes her message at rate h(q ∗ δ)− h(δ)
R1 = h(q ∗ δ)− h(δ) , R2 = R3 = 1− h(ǫ)
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Symmetry and addition saved the world
We have banked on
Broadcast and interference
![Page 86: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/86.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Symmetry and addition saved the world
We have banked on
Channels of Rec. 2 and 3 are symmetric
so uniform input distribution achieves capacity
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Symmetry and addition saved the world
We have banked on
Channels of Rec. 2 and 3 are symmetric
so uniform input distribution achieves capacity
Interference in the broadcast channel is additive
Broadcast and interference
![Page 88: New communication strategies for broadcast and interference … · Broadcast Communication Nested linear codes Groups and Nested Group Codes Applications Digital watermarking, data](https://reader034.vdocuments.us/reader034/viewer/2022052022/60378805326a904deb102485/html5/thumbnails/88.jpg)
Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Symmetry and addition saved the world
We have banked on
Channels of Rec. 2 and 3 are symmetric
so uniform input distribution achieves capacity
Interference in the broadcast channel is additive
But Shannon theory is all about not getting bogged down in an
example
Objective is to develop a theory for general case
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
However?
Caution: Even in point-to-point communication
In general, linear codes do not achieve Shannon capacity of an
arbitrary discrete memoryless channel
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
However?
Caution: Even in point-to-point communication
In general, linear codes do not achieve Shannon capacity of an
arbitrary discrete memoryless channel
What hope do we have in using them for network
communication for the arbitrary discrete memoryless case?
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Thesis
Algebraic structure in codes may be necessary in a
fundamental way
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Thesis
Algebraic structure in codes may be necessary in a
fundamental way
Algebraic structure alone is not sufficient
A right mix of algebraic structure along with non-linearity
Nested algebraic code appears to be a universal structure
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Noisy Channel Coding in Point-to-point case
ChannelYn
M MXn
Channel Encoder Channel Decoder
Given: Channel I/P= X , O/P=Y , with pY |X , and cost
function w(x)
Find: maximum transmission rate R for a target cost W .
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Noisy Channel Coding in Point-to-point case
ChannelYn
M MXn
Channel Encoder Channel Decoder
Given: Channel I/P= X , O/P=Y , with pY |X , and cost
function w(x)
Find: maximum transmission rate R for a target cost W .
Answer: Shannon Capacity-Cost function (Shannon ’49)
C (W ) = maxpX :Ew≤W
I (X ;Y )
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of a near-optimal channel code
Obtained from Shannon ensembleBox = X n
Red dot = codeword
C = code book
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of a near-optimal channel code
Obtained from Shannon ensembleBox = X n
Red dot = codeword
C = code book
C has Packing Property
C has Shaping Property
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Picture of a near-optimal channel code
Obtained from Shannon ensembleBox = X n
Red dot = codeword
C = code book
C has Packing Property
C has Shaping Property
Shape Region = Typical set
Size of code = I (X ;Y )
Codeword density =
I (X ;Y )− H(X ) = −H(X |Y )
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
New Result: An optimal linear code
Let |X | = p, prime no.
C1 = code book
C1 has Packing Property
Size of code
=log |X | − H(X |Y )
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
New Result: An optimal linear code
Let |X | = p, prime no.
C1 = code book
C1 has Packing Property
Size of code
=log |X | − H(X |Y )
Finite field is Zp
Bounding Region = X n
Density = −H(X |Y )
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
New Theorem: An optimal nested linear code
Going beyond symmetry
C1 fine code (red & black)
C2 coarse code (black)
C1 has Packing property
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
New Theorem: An optimal nested linear code
Going beyond symmetry
C1 fine code (red & black)
C2 coarse code (black)
C1 has Packing property
C2 has Shaping property
Size of C1 = log |X | − H(X |Y )
Size of C2 = log |X | − H(X )
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
New Theorem: An optimal nested linear code
Going beyond symmetry
C1 fine code (red & black)
C2 coarse code (black)
C1 has Packing property
C2 has Shaping property
Size of C1 = log |X | − H(X |Y )
Size of C2 = log |X | − H(X )
Code book= C1/C2
Code book size = I (X ;Y )
Achieves C (W )
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Going beyond addition
X2 ∨ X3 (logical OR function)
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Going beyond addition
X2 ∨ X3 (logical OR function)
What kind of glasses you wear so this looks like addition?
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Going beyond addition
X2 ∨ X3 (logical OR function)
What kind of glasses you wear so this looks like addition?
Can be embedded in the addition table in F3
0
1
2
10 2
0
0
0
1
1
1
2
2
2
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Going beyond addition
X2 ∨ X3 (logical OR function)
What kind of glasses you wear so this looks like addition?
Can be embedded in the addition table in F3
0
1
2
10 2
0
0
0
1
1
1
2
2
2
Map binary sources into F3, and use linear codes built on F3
Can do better than traditional random coding
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Going beyond addition
X2 ∨ X3 (logical OR function)
What kind of glasses you wear so this looks like addition?
Can be embedded in the addition table in F3
0
1
2
10 2
0
0
0
1
1
1
2
2
2
Map binary sources into F3, and use linear codes built on F3
Can do better than traditional random coding
In general we ‘embed’ bivariate functions in groups
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Groups - An Introduction
G - a finite abelian group of order n
G ∼= Zpe11× Zp
e22· · · × Z
pekk
G isomorphic to direct product of possibly repeating primary
cyclic groups
g ∈ G ⇔ g = (g1, . . . , gk), gi ∈ Zpeii
Call gi as the ith digit of g
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Groups - An Introduction
G - a finite abelian group of order n
G ∼= Zpe11× Zp
e22· · · × Z
pekk
G isomorphic to direct product of possibly repeating primary
cyclic groups
g ∈ G ⇔ g = (g1, . . . , gk), gi ∈ Zpeii
Call gi as the ith digit of g
Prove coding theorems for primary cyclic groups
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Nested Group Codes
Group code over Znpr : C < Z
npr
C = Image(φ) for some homomorphism φ : Zkpr → Z
npr
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Nested Group Codes
Group code over Znpr : C < Z
npr
C = Image(φ) for some homomorphism φ : Zkpr → Z
npr
(C1, C2) nested if C2 ⊂ C1
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Nested Group Codes
Group code over Znpr : C < Z
npr
C = Image(φ) for some homomorphism φ : Zkpr → Z
npr
(C1, C2) nested if C2 ⊂ C1
We need:
C1 < Znpr : “good” packing code
C2 < Znpr : “good” covering code
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Good Group Packing Codes
Good group channel code C2 for the triple (U ,V ,PUV )
Assume U = Zpr for some prime p and exponent r > 0
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Good Group Packing Codes
Good group channel code C2 for the triple (U ,V ,PUV )
Assume U = Zpr for some prime p and exponent r > 0
Lemma
Exists for large n if
1nlog |C2| ≤ log pr −max0≤i<r
(
rr−i
)
(H(U|V )− H([U]i |V ))
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Good Group Packing Codes
Good group channel code C2 for the triple (U ,V ,PUV )
Assume U = Zpr for some prime p and exponent r > 0
Lemma
Exists for large n if
1nlog |C2| ≤ log pr −max0≤i<r
(
rr−i
)
(H(U|V )− H([U]i |V ))
[U]i is a function of U and depends on the group
Extra penalty for imposing group structure beyond linearity
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Good Group Packing Codes
Good group channel code C2 for the triple (U ,V ,PUV )
Assume U = Zpr for some prime p and exponent r > 0
Lemma
Exists for large n if
1nlog |C2| ≤ log pr −max0≤i<r
(
rr−i
)
(H(U|V )− H([U]i |V ))
[U]i is a function of U and depends on the group
Extra penalty for imposing group structure beyond linearity
Time for questions?
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
A Distributed Source Coding Problem
Rate=R2Y
2
Encoder
Rate=R3Z
3
Encoder
Rate=R1X
1
Encoder
Decoder
JointX ,Y ,Z
n
n
n
n n n
^
^ ^ ^
Encoders observe different components of a vector source
Central decoder receives quantized observations from the
encoders
Given source distribution pXYZ
Best known rate region - Berger-Tung Rate Region, ’77
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Conclusions
Presented a nested group codes based coding scheme
Can recover known rate regions of broadcast channel
Offers rate gains over random coding coding scheme
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Conclusions
Presented a nested group codes based coding scheme
Can recover known rate regions of broadcast channel
Offers rate gains over random coding coding scheme
New bridge between probability and algebra, between
information theory and coding theory
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Conclusions
Presented a nested group codes based coding scheme
Can recover known rate regions of broadcast channel
Offers rate gains over random coding coding scheme
New bridge between probability and algebra, between
information theory and coding theory
It was thought that probability and algebra are nemesis
Broadcast and interference
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Introduction
Broadcast Communication
Nested linear codes
Groups and Nested Group Codes
Conclusions
Presented a nested group codes based coding scheme
Can recover known rate regions of broadcast channel
Offers rate gains over random coding coding scheme
New bridge between probability and algebra, between
information theory and coding theory
It was thought that probability and algebra are nemesis
Instead the match made in heaven
Broadcast and interference