Towards an Algebraic Network Information Theory Bobak Nazer (BU) - - PowerPoint PPT Presentation

Towards an Algebraic Network Information Theory

Bobak Nazer (BU) Joint work with Sung Hoon Lim (EPFL), Chen Feng (UBC), and Michael Gastpar (EPFL). DIMACS Workshop on Network Coding: The Next 15 Years December 17th, 2015

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations.

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Classical Approach:

• Generate codewords elementwise i.i.d.

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Classical Approach:

• Generate codewords elementwise i.i.d.
• Powerful generalizations including superposition coding, dirty paper coding,

block Markov coding, and many more...

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Classical Approach:

• Generate codewords elementwise i.i.d.
• Powerful generalizations including superposition coding, dirty paper coding,

block Markov coding, and many more...

• Rate regions described in terms of (single-letter) information measures
• ptimized over pmfs.

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Classical Approach:

• Generate codewords elementwise i.i.d.
• Powerful generalizations including superposition coding, dirty paper coding,

block Markov coding, and many more...

• Rate regions described in terms of (single-letter) information measures
• ptimized over pmfs.
• Many important successes: multiple-access channels, (degraded) broadcast

channels, Slepian-Wolf compression, network coding, and many more...

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Classical Approach:

• Generate codewords elementwise i.i.d.
• Powerful generalizations including superposition coding, dirty paper coding,

block Markov coding, and many more...

• Rate regions described in terms of (single-letter) information measures
• ptimized over pmfs.
• Many important successes: multiple-access channels, (degraded) broadcast

channels, Slepian-Wolf compression, network coding, and many more...

• State-of-the-art elegantly captured in the recent textbook of

El Gamal and Kim.

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Classical Approach:

• Generate codewords elementwise i.i.d.
• Powerful generalizations including superposition coding, dirty paper coding,

block Markov coding, and many more...

• Rate regions described in terms of (single-letter) information measures
• ptimized over pmfs.
• Many important successes: multiple-access channels, (degraded) broadcast

channels, Slepian-Wolf compression, network coding, and many more...

• State-of-the-art elegantly captured in the recent textbook of

El Gamal and Kim.

• Codes with algebraic structure are sought after to mimic the performance
• f random i.i.d. codes.

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations.

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Algebraic Approach:

• Utilize linear or lattice codebooks.

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Algebraic Approach:

• Utilize linear or lattice codebooks.
• Compelling examples starting from the work of K¨
• rner and Marton on

distributed compression and, more recently, many papers on physical-layer network coding, distributed dirty paper coding, and interference alignment.

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Algebraic Approach:

• Utilize linear or lattice codebooks.
• Compelling examples starting from the work of K¨
• rner and Marton on

distributed compression and, more recently, many papers on physical-layer network coding, distributed dirty paper coding, and interference alignment.

• Coding schemes exhibit behavior not found via i.i.d. ensembles.

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Algebraic Approach:

• Utilize linear or lattice codebooks.
• Compelling examples starting from the work of K¨
• rner and Marton on

distributed compression and, more recently, many papers on physical-layer network coding, distributed dirty paper coding, and interference alignment.

• Coding schemes exhibit behavior not found via i.i.d. ensembles.
• However, some classical coding techniques are still unavailable.

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Algebraic Approach:

• Utilize linear or lattice codebooks.
• Compelling examples starting from the work of K¨
• rner and Marton on

distributed compression and, more recently, many papers on physical-layer network coding, distributed dirty paper coding, and interference alignment.

• Coding schemes exhibit behavior not found via i.i.d. ensembles.
• However, some classical coding techniques are still unavailable.
• Most of the initial efforts have focused on Gaussian networks and have

employed nested lattice codebooks.

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Algebraic Approach:

• Utilize linear or lattice codebooks.
• Compelling examples starting from the work of K¨
• rner and Marton on

distributed compression and, more recently, many papers on physical-layer network coding, distributed dirty paper coding, and interference alignment.

• Coding schemes exhibit behavior not found via i.i.d. ensembles.
• However, some classical coding techniques are still unavailable.
• Most of the initial efforts have focused on Gaussian networks and have

employed nested lattice codebooks.

• Are these just a collection of intriguing examples or elements of a more

general theory?

Network Information Theory

Goal: Roughly speaking, for a given network, determine necessary and sufficient conditions on the rates at which the sources (or some functions thereof) can be communicated to the destinations. Algebraic Approach:

• Utilize linear or lattice codebooks.
• Compelling examples starting from the work of K¨
• rner and Marton on

distributed compression and, more recently, many papers on physical-layer network coding, distributed dirty paper coding, and interference alignment.

• Coding schemes exhibit behavior not found via i.i.d. ensembles.
• However, some classical coding techniques are still unavailable.
• Most of the initial efforts have focused on Gaussian networks and have

employed nested lattice codebooks.

• Are these just a collection of intriguing examples or elements of a more

general theory?

This Talk: We build on previous work and propose a joint typicality approach to algebraic network information theory.

Compute-and-Forward

Goal: Send a linear combination of the messages to the receiver.

Compute-and-Forward

Goal: Send a linear combination of the messages to the receiver.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . . Channel

Y n D ˆ t

Compute-and-Forward

Goal: Send a linear combination of the messages to the receiver.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . .

ν ν ν(·) = q-ary expansion

Fκ

q Channel

Y n D ˆ t ν ν ν(t) =

K

• k=1

ak ν ν ν(mk)

Compute-and-Forward

Goal: Send linear combinations of the messages to the receivers.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . .

ν ν ν(·) = q-ary expansion

Fκ

q Channel

Y n

1

D1 ˆ t1 Y n

2

D2 ˆ t2

. . .

. . . . . . Y n

K

DK ˆ tK ν ν ν(tℓ) =

K

• k=1

aℓ,k ν ν ν(mk)

Compute-and-Forward

Goal: Send linear combinations of the messages to the receivers.

• Compute-and-forward can serve as a framework for communicating

messages across a network (e.g., relaying, MIMO uplink/downlink, interference alignment).

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . .

ν ν ν(·) = q-ary expansion

Fκ

q Channel

Y n

1

D1 ˆ t1 Y n

2

D2 ˆ t2

. . .

. . . . . . Y n

K

DK ˆ tK ν ν ν(tℓ) =

K

• k=1

aℓ,k ν ν ν(mk)

Compute-and-Forward

Goal: Send linear combinations of the messages to the receivers.

• Compute-and-forward can serve as a framework for communicating

messages across a network (e.g., relaying, MIMO uplink/downlink, interference alignment).

• Much of the recent work has focused on Gaussian networks.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . .

Rn

ν ν ν(·) = q-ary expansion

Fκ

q

H

Zn

1

Zn

2

Zn

K

Y n

1

D1 ˆ t1 Y n

2

D2 ˆ t2

. . .

. . . . . . Y n

K

DK ˆ tK ν ν ν(tℓ) =

K

• k=1

aℓ,k ν ν ν(mk)

The Usual Approach

The Usual Approach

Computation over Gaussian MACs

• Symmetric Gaussian MAC.

Computation over Gaussian MACs

• Symmetric Gaussian MAC.
• Equal power constraints:

Exℓ2 ≤ nP.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . Zn Y n D ˆ t ν ν ν(t) =

K

• k=1

ν ν ν(mk)

Computation over Gaussian MACs

• Symmetric Gaussian MAC.
• Equal power constraints:

Exℓ2 ≤ nP.

• Use nested lattice codes.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . . . . . Zn Y n D ˆ t ν ν ν(t) =

K

• k=1

ν ν ν(mk)

Computation over Gaussian MACs

• Symmetric Gaussian MAC.
• Equal power constraints:

Exℓ2 ≤ nP.

• Use nested lattice codes.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . . . . . Zn Y n D ˆ t ν ν ν(t) =

K

• k=1

ν ν ν(mk)

• Wilson-Narayanan-Pfister-Sprintson ’10, Nazer-Gastpar ’11:

Decoding is successful if the rates satisfy Rk < 1 2 log+ 1 2 + P

• .

Computation over Gaussian MACs

• Symmetric Gaussian MAC.
• Equal power constraints:

Exℓ2 ≤ nP.

• Use nested lattice codes.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . . . . . Zn Y n D ˆ t ν ν ν(t) =

K

• k=1

ν ν ν(mk)

• Wilson-Narayanan-Pfister-Sprintson ’10, Nazer-Gastpar ’11:

Decoding is successful if the rates satisfy Rk < 1 2 log+ 1 2 + P

• .
• Cut-set upper bound is 1

2 log(1 + P).

Computation over Gaussian MACs

• Symmetric Gaussian MAC.
• Equal power constraints:

Exℓ2 ≤ nP.

• Use nested lattice codes.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . . . . . Zn Y n D ˆ t ν ν ν(t) =

K

• k=1

ν ν ν(mk)

• Wilson-Narayanan-Pfister-Sprintson ’10, Nazer-Gastpar ’11:

Decoding is successful if the rates satisfy Rk < 1 2 log+ 1 2 + P

• .
• Cut-set upper bound is 1

2 log(1 + P).

• What about the “1+”? Still open! (Ice wine problem.)

Computation over Gaussian MACs

• How about general

Gaussian MACs?

Computation over Gaussian MACs

• How about general

Gaussian MACs?

• Model using unequal

power constraints: Exℓ2 ≤ nPℓ.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . Zn Y n D ˆ t ν ν ν(t) =

K

• k=1
• 0 ν

ν ν(mk)

Computation over Gaussian MACs

• How about general

Gaussian MACs?

• Model using unequal

power constraints: Exℓ2 ≤ nPℓ.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . . . . . Zn Y n D ˆ t ν ν ν(t) =

K

• k=1
• 0 ν

ν ν(mk)

• Nam-Chung-Lee ’11: At each transmitter, use the same fine lattice

and a different coarse lattice, chosen to meet the power constraint.

Computation over Gaussian MACs

• How about general

Gaussian MACs?

• Model using unequal

power constraints: Exℓ2 ≤ nPℓ.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . . . . . Zn Y n D ˆ t ν ν ν(t) =

K

• k=1
• 0 ν

ν ν(mk)

• Nam-Chung-Lee ’11: At each transmitter, use the same fine lattice

and a different coarse lattice, chosen to meet the power constraint.

• Decoding is successful if the rates satisfy

Rℓ < 1 2 log+

• Pℓ

L

i=1 Pi

+ Pℓ

• .

Computation over Gaussian MACs

• How about general

Gaussian MACs?

• Model using unequal

power constraints: Exℓ2 ≤ nPℓ.

m1 E1 Xn

1

m2 E2 Xn

2

mK EK Xn

K

. . . . . . . . . . . . Zn Y n D ˆ t ν ν ν(t) =

K

• k=1
• 0 ν

ν ν(mk)

• Nam-Chung-Lee ’11: At each transmitter, use the same fine lattice

and a different coarse lattice, chosen to meet the power constraint.

• Decoding is successful if the rates satisfy

Rℓ < 1 2 log+

• Pℓ

L

i=1 Pi

+ Pℓ

• .
• Nazer-Cadambe-Ntranos-Caire ’15: Expanded compute-and-forward

framework to link unequal power setting to finite fields.

Point-to-Point Channels

M

Encoder

Xn pY |X Y n

Decoder

ˆ M

• Messages: m ∈ [2nR] {0, . . . , 2nR − 1}
• Encoder: a mapping xn(m) ∈ X n for each m ∈ [2nR]
• Decoder: a mapping ˆ

m(yn) ∈ [2nR] for each yn ∈ Yn

Point-to-Point Channels

M

Encoder

Xn pY |X Y n

Decoder

ˆ M

• Messages: m ∈ [2nR] {0, . . . , 2nR − 1}
• Encoder: a mapping xn(m) ∈ X n for each m ∈ [2nR]
• Decoder: a mapping ˆ

m(yn) ∈ [2nR] for each yn ∈ Yn

Theorem (Shannon ’48)

C = max

pX(x) I(X; Y )

Point-to-Point Channels

M

Encoder

Xn pY |X Y n

Decoder

ˆ M

• Messages: m ∈ [2nR] {0, . . . , 2nR − 1}
• Encoder: a mapping xn(m) ∈ X n for each m ∈ [2nR]
• Decoder: a mapping ˆ

m(yn) ∈ [2nR] for each yn ∈ Yn

Theorem (Shannon ’48)

C = max

pX(x) I(X; Y )

• Proof relies on random i.i.d. codebooks combined with

joint typicality decoding.

Random i.i.d. Codebooks

X n

T (n)

ǫ

(X)

Random i.i.d. Codes

• Codewords are independent of one another.
• Can directly target an input distribution pX(x).

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Pick a finite field Fq and a symbol mapping x : Fq → X.

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Pick a finite field Fq and a symbol mapping x : Fq → X.
• Set κ = nR/ log(q).

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Pick a finite field Fq and a symbol mapping x : Fq → X.
• Set κ = nR/ log(q).
• Draw a random generator matrix G ∈ Fκ×n

q

elementwise i.i.d. Unif(Fq). Let G be a realization.

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Pick a finite field Fq and a symbol mapping x : Fq → X.
• Set κ = nR/ log(q).
• Draw a random generator matrix G ∈ Fκ×n

q

elementwise i.i.d. Unif(Fq). Let G be a realization.

• Draw a random shift (or “dither”) Dn elementwise i.i.d. Unif(Fq).

Let dn be a realization.

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Pick a finite field Fq and a symbol mapping x : Fq → X.
• Set κ = nR/ log(q).
• Draw a random generator matrix G ∈ Fκ×n

q

elementwise i.i.d. Unif(Fq). Let G be a realization.

• Draw a random shift (or “dither”) Dn elementwise i.i.d. Unif(Fq).

Let dn be a realization.

• Take q-ary expansion of message m into the vector ν

ν ν(m) ∈ Fκ

q.

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Pick a finite field Fq and a symbol mapping x : Fq → X.
• Set κ = nR/ log(q).
• Draw a random generator matrix G ∈ Fκ×n

q

elementwise i.i.d. Unif(Fq). Let G be a realization.

• Draw a random shift (or “dither”) Dn elementwise i.i.d. Unif(Fq).

Let dn be a realization.

• Take q-ary expansion of message m into the vector ν

ν ν(m) ∈ Fκ

q.

• Linear codeword for message m is un(m) = ν

ν ν(m)G ⊕ dn.

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Pick a finite field Fq and a symbol mapping x : Fq → X.
• Set κ = nR/ log(q).
• Draw a random generator matrix G ∈ Fκ×n

q

elementwise i.i.d. Unif(Fq). Let G be a realization.

• Draw a random shift (or “dither”) Dn elementwise i.i.d. Unif(Fq).

Let dn be a realization.

• Take q-ary expansion of message m into the vector ν

ν ν(m) ∈ Fκ

q.

• Linear codeword for message m is un(m) = ν

ν ν(m)G ⊕ dn.

• Channel input at time i is xi(m) = x(ui(m)).

Random i.i.d. Codebooks

Random Linear Codes

X n

T (n)

ǫ

(X)

• Codewords are pairwise independent of one another.
• Codewords are uniformly distributed over Fn

q.

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M

• Well known that a direct application of linear coding is not sufficient

to reach the point-to-point capacity, Ahlswede ’71.

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M

• Well known that a direct application of linear coding is not sufficient

to reach the point-to-point capacity, Ahlswede ’71.

• Gallager ’68: Pick Fq with q ≫ X and choose symbol mapping x(u)

to reach c.a.i.d. from Unif(Fq). This can attain the capacity.

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M

• Well known that a direct application of linear coding is not sufficient

to reach the point-to-point capacity, Ahlswede ’71.

• Gallager ’68: Pick Fq with q ≫ X and choose symbol mapping x(u)

to reach c.a.i.d. from Unif(Fq). This can attain the capacity.

• This will not work for us. Roughly speaking, if each encoder has a

different input distribution, the symbol mappings may be quite different, which will disrupt the linear structure of the codebook.

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M

• Well known that a direct application of linear coding is not sufficient

to reach the point-to-point capacity, Ahlswede ’71.

• Gallager ’68: Pick Fq with q ≫ X and choose symbol mapping x(u)

to reach c.a.i.d. from Unif(Fq). This can attain the capacity.

• This will not work for us. Roughly speaking, if each encoder has a

different input distribution, the symbol mappings may be quite different, which will disrupt the linear structure of the codebook.

• Padakandla-Pradhan ’13: It is possible to shape the input

distribution using nested linear codes.

Point-to-Point Channels: Linear Codes

M

Linear Code

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M

• Well known that a direct application of linear coding is not sufficient

to reach the point-to-point capacity, Ahlswede ’71.

• Gallager ’68: Pick Fq with q ≫ X and choose symbol mapping x(u)

to reach c.a.i.d. from Unif(Fq). This can attain the capacity.

• This will not work for us. Roughly speaking, if each encoder has a

different input distribution, the symbol mappings may be quite different, which will disrupt the linear structure of the codebook.

• Padakandla-Pradhan ’13: It is possible to shape the input

distribution using nested linear codes.

• Basic idea: Generate many codewords to represent one message.

Search in this “bin” to find a codeword with the desired type, i.e., multicoding.

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Messages m ∈ [2nR] and auxiliary indices l ∈ [2n ˆ

R].

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Messages m ∈ [2nR] and auxiliary indices l ∈ [2n ˆ

R].

• Set κ = n(R + ˆ

R)/ log(q).

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Messages m ∈ [2nR] and auxiliary indices l ∈ [2n ˆ

R].

• Set κ = n(R + ˆ

R)/ log(q).

• Pick generator matrix G and dither dn as before.

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Messages m ∈ [2nR] and auxiliary indices l ∈ [2n ˆ

R].

• Set κ = n(R + ˆ

R)/ log(q).

• Pick generator matrix G and dither dn as before.
• Take q-ary expansions
• ν

ν ν(m) ν ν ν(l)

• ∈ Fκ

q.

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Code Construction:

• Messages m ∈ [2nR] and auxiliary indices l ∈ [2n ˆ

R].

• Set κ = n(R + ˆ

R)/ log(q).

• Pick generator matrix G and dither dn as before.
• Take q-ary expansions
• ν

ν ν(m) ν ν ν(l)

• ∈ Fκ

q.

• Linear codewords: un(m, l) =
• ν

ν ν(m) ν ν ν(l)

• G ⊕ dn.

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Encoding:

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Encoding:

• Fix p(u) and x(u).

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Encoding:

• Fix p(u) and x(u).
• Multicoding: For each m, find an index l such that

un(m, l) ∈ T (n)

ǫ′

(U)

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Encoding:

• Fix p(u) and x(u).
• Multicoding: For each m, find an index l such that

un(m, l) ∈ T (n)

ǫ′

(U)

• Succeeds w.h.p. if ˆ

R > D(pUpq) (where pq is uniform over Fq).

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Encoding:

• Fix p(u) and x(u).
• Multicoding: For each m, find an index l such that

un(m, l) ∈ T (n)

ǫ′

(U)

• Succeeds w.h.p. if ˆ

R > D(pUpq) (where pq is uniform over Fq).

• Transmit xi = x
• ui(m, l)
• .

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Encoding:

• Fix p(u) and x(u).
• Multicoding: For each m, find an index l such that

un(m, l) ∈ T (n)

ǫ′

(U)

• Succeeds w.h.p. if ˆ

R > D(pUpq) (where pq is uniform over Fq).

• Transmit xi = x
• ui(m, l)
• .

Decoding:

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Encoding:

• Fix p(u) and x(u).
• Multicoding: For each m, find an index l such that

un(m, l) ∈ T (n)

ǫ′

(U)

• Succeeds w.h.p. if ˆ

R > D(pUpq) (where pq is uniform over Fq).

• Transmit xi = x
• ui(m, l)
• .

Decoding:

• Joint Typicality Decoding: Find the unique index ˆ

m such that

• un( ˆ

m, ˆ l), yn) ∈ T (n)

ǫ

(U, Y ) for some index ˆ l.

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M Encoding:

• Fix p(u) and x(u).
• Multicoding: For each m, find an index l such that

un(m, l) ∈ T (n)

ǫ′

(U)

• Succeeds w.h.p. if ˆ

R > D(pUpq) (where pq is uniform over Fq).

• Transmit xi = x
• ui(m, l)
• .

Decoding:

• Joint Typicality Decoding: Find the unique index ˆ

m such that

• un( ˆ

m, ˆ l), yn) ∈ T (n)

ǫ

(U, Y ) for some index ˆ l.

• Succeeds w.h.p. if R + ˆ

R < I(U; Y ) + D(pUpq)

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M

Theorem (Padakandla-Pradhan ’13)

Any rate R satisfying R < max

p(u), x(u) I(U; Y )

is achievable. This is equal to the capacity if q ≥ |X|.

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M

Theorem (Padakandla-Pradhan ’13)

Any rate R satisfying R < max

p(u), x(u) I(U; Y )

is achievable. This is equal to the capacity if q ≥ |X|.

• This is the basic coding framework that we will use for each

transmitter.

Point-to-Point Channels: Linear Codes + Multicoding

M

Linear Code Multi- coding

U n x(u)

Encoder

Xn pY |X Y n

Decoder

ˆ M

Theorem (Padakandla-Pradhan ’13)

Any rate R satisfying R < max

p(u), x(u) I(U; Y )

is achievable. This is equal to the capacity if q ≥ |X|.

• This is the basic coding framework that we will use for each

transmitter.

• Next, let’s examine a two-transmitter, one-receiver

“compute-and-forward” network.

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Code Construction:

• Messages mk ∈ [2nRk] and auxiliary indices lk ∈ [2n ˆ

Rk], k = 1, 2.

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Code Construction:

• Messages mk ∈ [2nRk] and auxiliary indices lk ∈ [2n ˆ

Rk], k = 1, 2.

• Set κ = n(max{R1 + ˆ

R1, R2 + ˆ R2})/ log(q).

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Code Construction:

• Messages mk ∈ [2nRk] and auxiliary indices lk ∈ [2n ˆ

Rk], k = 1, 2.

• Set κ = n(max{R1 + ˆ

R1, R2 + ˆ R2})/ log(q).

• Pick generator matrix G and dithers dn

1, dn 2 as before.

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Code Construction:

• Messages mk ∈ [2nRk] and auxiliary indices lk ∈ [2n ˆ

Rk], k = 1, 2.

• Set κ = n(max{R1 + ˆ

R1, R2 + ˆ R2})/ log(q).

• Pick generator matrix G and dithers dn

1, dn 2 as before.

• Take q-ary expansions
• ν

ν ν(m1) ν ν ν(l1)

• ∈ Fκ

q

• ν

ν ν(m2) ν ν ν(l2) 0

• ∈ Fκ

q

Zero-padding

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Code Construction:

• Messages mk ∈ [2nRk] and auxiliary indices lk ∈ [2n ˆ

Rk], k = 1, 2.

• Set κ = n(max{R1 + ˆ

R1, R2 + ˆ R2})/ log(q).

• Pick generator matrix G and dithers dn

1, dn 2 as before.

• Take q-ary expansions
• η

η η(m1, l1)

• ∈ Fκ

q

• η

η η(m2, l2)

• ∈ Fκ

q

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Code Construction:

• Messages mk ∈ [2nRk] and auxiliary indices lk ∈ [2n ˆ

Rk], k = 1, 2.

• Set κ = n(max{R1 + ˆ

R1, R2 + ˆ R2})/ log(q).

• Pick generator matrix G and dithers dn

1, dn 2 as before.

• Take q-ary expansions
• η

η η(m1, l1)

• ∈ Fκ

q

• η

η η(m2, l2)

• ∈ Fκ

q

• Linear codewords: un

1(m1, l1) = η

η η(m1, l1)G ⊕ dn

1

un

2(m2, l2) = η

η η(m2, l2)G ⊕ dn

2

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Encoding:

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Encoding:

• Fix p(u1), p(u2), x1(u1), and x2(u2).

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Encoding:

• Fix p(u1), p(u2), x1(u1), and x2(u2).
• Multicoding: For each mk, find an index lk such that

un

k(mk, lk) ∈ T (n) ǫ′

(Uk).

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Encoding:

• Fix p(u1), p(u2), x1(u1), and x2(u2).
• Multicoding: For each mk, find an index lk such that

un

k(mk, lk) ∈ T (n) ǫ′

(Uk).

• Succeeds w.h.p. if ˆ

Rk > D(pUkpq).

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Encoding:

• Fix p(u1), p(u2), x1(u1), and x2(u2).
• Multicoding: For each mk, find an index lk such that

un

k(mk, lk) ∈ T (n) ǫ′

(Uk).

• Succeeds w.h.p. if ˆ

Rk > D(pUkpq).

• Transmit xki = xk
• uki(mk, lk)
• .

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Encoding:

• Fix p(u1), p(u2), x1(u1), and x2(u2).
• Multicoding: For each mk, find an index lk such that

un

k(mk, lk) ∈ T (n) ǫ′

(Uk).

• Succeeds w.h.p. if ˆ

Rk > D(pUkpq).

• Transmit xki = xk
• uki(mk, lk)
• .

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Computation Problem:

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Computation Problem:

• Consider the coefficients a ∈ F2

q, a = [a1, a2]

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Computation Problem:

• Consider the coefficients a ∈ F2

q, a = [a1, a2]

• For mk ∈ [2nRk], lk ∈ [2n ˆ

Rk], the linear combination of codewords

with coefficient vector a is a1un

1(m1, l1) ⊕ a2un 2(m2, l2)

=

• a1η

η η(m1, l1) ⊕ a2η η η(m2, l2)

• G ⊕ a1dn

1 ⊕ a2dn 2

= ν ν ν(t)G ⊕ dn

w

= wn(t), t ∈ [2n max{R1+ ˆ

R1,R2+ ˆ R2}]

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Computation Problem:

• Let Mk be the chosen message and Lk the chosen index from the

multicoding step.

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Computation Problem:

• Let Mk be the chosen message and Lk the chosen index from the

multicoding step.

• Decoder wants a linear combination of the codewords:

W n(T) = a1U n

1 (M1, L1) ⊕ a2U n 2 (M2, L2)

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Computation Problem:

• Let Mk be the chosen message and Lk the chosen index from the

multicoding step.

• Decoder wants a linear combination of the codewords:

W n(T) = a1U n

1 (M1, L1) ⊕ a2U n 2 (M2, L2)

• Decoder: ˆ

t(yn) ∈ [2n max{R1+ ˆ

R1,R2+ ˆ R2}], yn ∈ Yn

• Probability of Error: P(n)

ǫ

= P{T = ˆ T}

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Computation Problem:

• Let Mk be the chosen message and Lk the chosen index from the

multicoding step.

• Decoder wants a linear combination of the codewords:

W n(T) = a1U n

1 (M1, L1) ⊕ a2U n 2 (M2, L2)

• Decoder: ˆ

t(yn) ∈ [2n max{R1+ ˆ

R1,R2+ ˆ R2}], yn ∈ Yn

• Probability of Error: P(n)

ǫ

= P{T = ˆ T}

• A rate pair is achievable if there exists a sequence of codes such that

P(n)

ǫ

→ 0 as n → ∞.

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T Decoding:

• Joint Typicality Decoding: Find an index t ∈ [2n max(R1+ ˆ

R1,R2+ ˆ R2)]

such that (wn(t), yn) ∈ T (n)

ǫ

.

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T

Theorem (Lim-Chen-Nazer-Gastpar Allerton ’15)

A rate pair (R1, R2) is achievable if R1 < I(W; Y ) − I(W; U2), R2 < I(W; Y ) − I(W; U1), for some p(u1)p(u2) and functions x1(u1), x2(u2), where Uk = Fq, k = 1, 2, and W = a1U1 ⊕ a2U2.

Nested Linear Coding Architecture

M1

Linear Code Multi- coding

U n

1

x1(u1) Xn

1

M2

Linear Code Multi- coding

U n

2

x2(u2) Xn

2

pY |X1X2 Y n

Decoder

ˆ T

Theorem (Lim-Chen-Nazer-Gastpar Allerton ’15)

A rate pair (R1, R2) is achievable if R1 < I(W; Y ) − I(W; U2), R2 < I(W; Y ) − I(W; U1), for some p(u1)p(u2) and functions x1(u1), x2(u2), where Uk = Fq, k = 1, 2, and W = a1U1 ⊕ a2U2.

• Padakandla-Pradhan ’13: Special case where R1 = R2.

Proof Sketch

• WLOG assume M = {M1 = 0, M2 = 0, L1 = 0, L2 = 0}.

Proof Sketch

• WLOG assume M = {M1 = 0, M2 = 0, L1 = 0, L2 = 0}.
• Union bound: P(n)

ǫ

≤

• t=0

P

• (W n(t), Y n) ∈ T (n)

ǫ

|M

• .

Proof Sketch

• WLOG assume M = {M1 = 0, M2 = 0, L1 = 0, L2 = 0}.
• Union bound: P(n)

ǫ

≤

• t=0

P

• (W n(t), Y n) ∈ T (n)

ǫ

|M

• .
• Notice that the Lk depend on the codebook so Y n and W n(t) are

not independent.

Proof Sketch

• WLOG assume M = {M1 = 0, M2 = 0, L1 = 0, L2 = 0}.
• Union bound: P(n)

ǫ

≤

• t=0

P

• (W n(t), Y n) ∈ T (n)

ǫ

|M

• .
• Notice that the Lk depend on the codebook so Y n and W n(t) are

not independent.

• To get around this issue, we analyze

P(E) =

• t=0

P

• (W n(t), Y n) ∈ T (n)

ǫ

, U n

1 (0, 0) ∈ T (n) ǫ

, U n

2 (0, 0) ∈ T (n) ǫ

|M

Proof Sketch

• WLOG assume M = {M1 = 0, M2 = 0, L1 = 0, L2 = 0}.
• Union bound: P(n)

ǫ

≤

• t=0

P

• (W n(t), Y n) ∈ T (n)

ǫ

|M

• .
• Notice that the Lk depend on the codebook so Y n and W n(t) are

not independent.

• To get around this issue, we analyze

P(E) =

• t=0

P

• (W n(t), Y n) ∈ T (n)

ǫ

, U n

1 (0, 0) ∈ T (n) ǫ

, U n

2 (0, 0) ∈ T (n) ǫ

|M

• Conditioned on M, Y n → (U n

1 (0, 0), U n 2 (0, 0)) → W n(t)

Proof Sketch

• WLOG assume M = {M1 = 0, M2 = 0, L1 = 0, L2 = 0}.
• Union bound: P(n)

ǫ

≤

• t=0

P

• (W n(t), Y n) ∈ T (n)

ǫ

|M

• .
• Notice that the Lk depend on the codebook so Y n and W n(t) are

not independent.

• To get around this issue, we analyze

P(E) =

• t=0

P

• (W n(t), Y n) ∈ T (n)

ǫ

, U n

1 (0, 0) ∈ T (n) ǫ

, U n

2 (0, 0) ∈ T (n) ǫ

|M

• Conditioned on M, Y n → (U n

1 (0, 0), U n 2 (0, 0)) → W n(t)

• P(E) tends to zero as n → ∞ if

Rk + ˆ Rk + ˆ R1 + ˆ R2 < I(W; Y ) + D(pW||pq) + D(pU1||pq) + D(pU2||pq)

Compute-and-Forward over a Gaussian MAC

• Consider a Gaussian MAC with real-valued channel output

Y = h1X1 + h2X2 + Z

Compute-and-Forward over a Gaussian MAC

• Consider a Gaussian MAC with real-valued channel output

Y = h1X1 + h2X2 + Z

• Want to recover a1Xn

1 + a2Xn 2 for some integers a1, a2.

Compute-and-Forward over a Gaussian MAC

• Consider a Gaussian MAC with real-valued channel output

Y = h1X1 + h2X2 + Z

• Want to recover a1Xn

1 + a2Xn 2 for some integers a1, a2.

• Gaussian noise: Z ∼ N(0, 1)

Compute-and-Forward over a Gaussian MAC

• Consider a Gaussian MAC with real-valued channel output

Y = h1X1 + h2X2 + Z

• Want to recover a1Xn

1 + a2Xn 2 for some integers a1, a2.

• Gaussian noise: Z ∼ N(0, 1)
• Usual power constraint: E[X2

k] ≤ P

Compute-and-Forward over a Gaussian MAC

• Consider a Gaussian MAC with real-valued channel output

Y = h1X1 + h2X2 + Z

• Want to recover a1Xn

1 + a2Xn 2 for some integers a1, a2.

• Gaussian noise: Z ∼ N(0, 1)
• Usual power constraint: E[X2

k] ≤ P

• Via Gaussian quantization arguments, we can recover the following

theorem.

Compute-and-Forward over a Gaussian MAC

• Consider a Gaussian MAC with real-valued channel output

Y = h1X1 + h2X2 + Z

• Want to recover a1Xn

1 + a2Xn 2 for some integers a1, a2.

• Gaussian noise: Z ∼ N(0, 1)
• Usual power constraint: E[X2

k] ≤ P

• Via Gaussian quantization arguments, we can recover the following

theorem.

Theorem (Nazer-Gastpar ’11)

For any channel vector h and integer coefficient vector a, any rate tuple satisfying Rk < Rcomp(h, a) for k s.t. ak = 0 is achievable where Rcomp(h, a) = 1 2 log+

• P

aT P −1I + hhT−1a

Beyond One Linear Combination

• In some scenarios, it is of interest to decode two or more linear

combinations at each receiver.

Beyond One Linear Combination

• In some scenarios, it is of interest to decode two or more linear

combinations at each receiver.

• For example, Ordentlich-Erez-Nazer ’14 approximates the sum

capacity of the symmetric Gaussian interference channel via decoding two linear combinations.

Beyond One Linear Combination

• In some scenarios, it is of interest to decode two or more linear

combinations at each receiver.

• For example, Ordentlich-Erez-Nazer ’14 approximates the sum

capacity of the symmetric Gaussian interference channel via decoding two linear combinations.

• Ordentlich-Erez-Nazer ’13 improves upon compute-and-forward for

two or more linear combinations via successive cancellation.

Beyond One Linear Combination

• In some scenarios, it is of interest to decode two or more linear

combinations at each receiver.

• For example, Ordentlich-Erez-Nazer ’14 approximates the sum

capacity of the symmetric Gaussian interference channel via decoding two linear combinations.

• Ordentlich-Erez-Nazer ’13 improves upon compute-and-forward for

two or more linear combinations via successive cancellation.

• What about jointly decoding the linear combinations?

Beyond One Linear Combination

• In some scenarios, it is of interest to decode two or more linear

combinations at each receiver.

• For example, Ordentlich-Erez-Nazer ’14 approximates the sum

capacity of the symmetric Gaussian interference channel via decoding two linear combinations.

• Ordentlich-Erez-Nazer ’13 improves upon compute-and-forward for

two or more linear combinations via successive cancellation.

• What about jointly decoding the linear combinations?
• Ordentlich-Erez ’13 derived bounds for lattice-based codes.

Beyond One Linear Combination

• In some scenarios, it is of interest to decode two or more linear

combinations at each receiver.

• For example, Ordentlich-Erez-Nazer ’14 approximates the sum

capacity of the symmetric Gaussian interference channel via decoding two linear combinations.

• Ordentlich-Erez-Nazer ’13 improves upon compute-and-forward for

two or more linear combinations via successive cancellation.

• What about jointly decoding the linear combinations?
• Ordentlich-Erez ’13 derived bounds for lattice-based codes.
• This talk: We can analyze this via joint typicality decoding to get an

achievable rate region.

Jointly Decoding Two Linear Combinations of K Codewords

• At node k ∈ [1 : K], the message Mk is encoded using the nested

linear coding architecture.

Jointly Decoding Two Linear Combinations of K Codewords

• At node k ∈ [1 : K], the message Mk is encoded using the nested

linear coding architecture.

• Let Lk be the chosen index from the multicoding step.

Jointly Decoding Two Linear Combinations of K Codewords

• At node k ∈ [1 : K], the message Mk is encoded using the nested

linear coding architecture.

• Let Lk be the chosen index from the multicoding step.
• The objective of the receiver is to compute two linear combinations
• f the codewords,

W n

1 (T1) = K

• k=1

a1kun

k(Mk, Lk)

W n

2 (T2) = K

• k=1

a2kun

k(Mk, Lk) ,

with vanishing probability of error.

Jointly Decoding Two Linear Combinations of K Codewords

• At node k ∈ [1 : K], the message Mk is encoded using the nested

linear coding architecture.

• Let Lk be the chosen index from the multicoding step.
• The objective of the receiver is to compute two linear combinations
• f the codewords,

W n

1 (T1) = K

• k=1

a1kun

k(Mk, Lk)

W n

2 (T2) = K

• k=1

a2kun

k(Mk, Lk) ,

with vanishing probability of error.

• Key Technical Issue: Random linear codewords are pairwise

independent, but not 4-wise independent!

Jointly Decoding Two Linear Combinations of K Codewords

Theorem (Lim-Chen-Nazer-Gastpar Allerton ’15)

A rate tuple (R1, . . . , RK) is achievable for computing two linear combinations if

Rk < min{H(Uk) − H(V |Y ), H(Uk) − H(W1, W2|Y, V )}, k ∈ K1 Rj < I(W2; Y, W1) − H(W2) + H(Uj), j ∈ K2, Rk + Rj < I(W1, W2; Y ) − H(W1, W2) + H(Uk) + H(Uj), k ∈ K1, j ∈ K2

• r

Rk < I(W1; Y, W2) − H(W1) + H(Uk), k ∈ K1, Rj < min{H(Uj) − H(V |Y ), H(Uj) − H(W1, W2|Y, V )}, j ∈ K2, Rk + Rj < I(W1, W2; Y ) − H(W1, W2) + H(Uk) + H(Uj), k ∈ K1, j ∈ K2

for some K

k=1 p(uk) and xk(uk) and non-zero vector b

b b ∈ F2

q,

where Kj = {k ∈ [1 : K] : ajk = 0}, j = 1, 2 and V = b1W1 ⊕ b2W2.

Jointly Decoding Two Linear Combinations of K Codewords

Theorem (Lim-Chen-Nazer-Gastpar Allerton ’15)

A rate tuple (R1, . . . , RK) is achievable for computing two linear combinations if

Rk < min{H(Uk) − H(V |Y ), H(Uk) − H(W1, W2|Y, V )}, k ∈ K1 Rj < I(W2; Y, W1) − H(W2) + H(Uj), j ∈ K2, Rk + Rj < I(W1, W2; Y ) − H(W1, W2) + H(Uk) + H(Uj), k ∈ K1, j ∈ K2

• r

Rk < I(W1; Y, W2) − H(W1) + H(Uk), k ∈ K1, Rj < min{H(Uj) − H(V |Y ), H(Uj) − H(W1, W2|Y, V )}, j ∈ K2, Rk + Rj < I(W1, W2; Y ) − H(W1, W2) + H(Uk) + H(Uj), k ∈ K1, j ∈ K2

for some K

k=1 p(uk) and xk(uk) and non-zero vector b

b b ∈ F2

q,

where Kj = {k ∈ [1 : K] : ajk = 0}, j = 1, 2 and V = b1W1 ⊕ b2W2.

• The auxiliary linear combination V plays a key role in classifying

dependent competing pairs in the error analysis.

Multiple-Access via Nested Linear Codes

Theorem (Lim-Chen-Nazer-Gastpar Allerton ’15)

A rate pair (R1, R2) is achievable for the discrete memoryless multiple-access channel if R1< max

a=0 min{H(U1) − H(W|Y ), H(U1) − H(U1, U2|Y, W)},

R2 < I(X2; Y |X1), R1 + R2 < I(X1, X2; Y ),

• r

R1 < I(X1; Y |X2), R2< max

a=0 min{H(U2) − H(W|Y ), H(U2) − H(U1, U2|Y, W)},

R1 + R2 < I(X1, X2; Y ) for some p(u1)p(u2) and x1(u1), x2(u2), where W = a1U1 ⊕ a2U2.

Multiple-Access Rate Region

R2 R1 R1 I1

R1 < I1, R2 < I(X2; Y |X1), R1 + R2 < I(X1, X2; Y ), where I1 = max

a=0 min{H(U1) − H(W|Y ), H(U1) − H(U1, U2|Y, W)}

Multiple-Access Rate Region

R2 I2 R1 R2

R1 < I(X1; Y |X2), R2 < I2, R1 + R2 < I(X1, X2; Y ), where I2 = max

a=0 min{H(U2) − H(W|Y ), H(U2) − H(U1, U2|Y, W)}

Multiple-Access Rate Region

R2 I2 R1 I1

• Multiple-access rate region via nested linear codes:

R1 ∪ R2

“Two Help One”

R2 R1 MAC Capacity Region

• Even if the receiver is only

interested in recovering one linear combination it can sometimes help to decode two!

“Two Help One”

R2 CF2 R1 CF1 Decode One Linear Combination

• Even if the receiver is only

interested in recovering one linear combination it can sometimes help to decode two!

“Two Help One”

R2 CF2 R1 CF1 Multiple-Access via Nested Linear Codes

• Even if the receiver is only

interested in recovering one linear combination it can sometimes help to decode two!

“Two Help One”

R2 CF2 R1 CF1 Union

• Even if the receiver is only

interested in recovering one linear combination it can sometimes help to decode two!

Case Study: Two-Sender, Two-Receiver Network

M1 E1 Xn

1

M2 E2 Xn

2

1 √ 2 1 1

Zn

1

Y n

1

Zn

2

Y n

2

D1 ( ˆ M1, ˆ M2) D2 Xn

1 + Xn 2

Case Study: Two-Sender, Two-Receiver Network

R1 R2 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 MAC capacity 1 MAC capacity 2

Case Study: Two-Sender, Two-Receiver Network

R1 R2 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 Nested linear codes 2

Case Study: Two-Sender, Two-Receiver Network

R1 R2 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 Nested linear codes 1

Case Study: Two-Sender, Two-Receiver Network

R1 R2 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 Nested linear codes Lattice based SC-CF Union of MACs

Concluding Remarks

• First steps towards bringing algebraic network information theory

back into the realm of joint typicality.

• Joint decoding rate region for compute-and-forward that
• utperforms parallel and successive decoding.