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

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). X n Y n 1 1 m 1 ˆ E 1 D 1 t 1 X n Y n 2 2 ˆ m 2 E 2 D 2 t 2 ν ν ( · ) = q-ary expansion ν Channel K . . . . . . � . . . . ν ν ν ( t ℓ ) = a ℓ,k ν ν ν ( m k ) . . . . . . . . k =1 X n Y n K K ˆ m K E K D K t K F κ q

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. Z n 1 X n Y n 1 1 m 1 ˆ E 1 D 1 t 1 Z n 2 X n Y n 2 H 2 ˆ m 2 E 2 D 2 t 2 ν ν ν ( · ) = q-ary expansion K . . . . . . � . . . . ν ν ν ( t ℓ ) = a ℓ,k ν ν ν ( m k ) . . . . . . . . Z n K k =1 X n Y n K K ˆ m K E K D K t K F κ R n q

The Usual Approach

The Usual Approach

Computation over Gaussian MACs • Symmetric Gaussian MAC.

Computation over Gaussian MACs • Symmetric Gaussian MAC. X n m 1 1 E 1 Z n • Equal power constraints: E � x ℓ � 2 ≤ nP . X n Y n m 2 2 ˆ E 2 D t . . . . K . . � ν ν ν ( t ) = ν ν ν ( m k ) X n K m K E K k =1

Computation over Gaussian MACs • Symmetric Gaussian MAC. X n m 1 1 E 1 Z n • Equal power constraints: E � x ℓ � 2 ≤ nP . X n Y n m 2 2 ˆ E 2 D t • Use nested lattice codes. . . . . . . . . K . . . . � ν ν ν ( t ) = ν ν ν ( m k ) X n K m K E K k =1

Computation over Gaussian MACs • Symmetric Gaussian MAC. X n m 1 1 E 1 Z n • Equal power constraints: E � x ℓ � 2 ≤ nP . X n Y n m 2 2 ˆ E 2 D t • Use nested lattice codes. . . . . . . . . K . . . . � ν ν ν ( t ) = ν ( m k ) ν ν X n K m K E K k =1 • Wilson-Narayanan-Pfister-Sprintson ’10, Nazer-Gastpar ’11: Decoding is successful if the rates satisfy � 1 � R k < 1 2 log + 2 + P .

Computation over Gaussian MACs • Symmetric Gaussian MAC. X n m 1 1 E 1 Z n • Equal power constraints: E � x ℓ � 2 ≤ nP . X n Y n m 2 2 ˆ E 2 D t • Use nested lattice codes. . . . . . . . . K . . . . � ν ν ν ( t ) = ν ( m k ) ν ν X n K m K E K k =1 • Wilson-Narayanan-Pfister-Sprintson ’10, Nazer-Gastpar ’11: Decoding is successful if the rates satisfy � 1 � R k < 1 2 log + 2 + P . • Cut-set upper bound is 1 2 log(1 + P ) .

Computation over Gaussian MACs • Symmetric Gaussian MAC. X n m 1 1 E 1 Z n • Equal power constraints: E � x ℓ � 2 ≤ nP . X n Y n m 2 2 ˆ E 2 D t • Use nested lattice codes. . . . . . . . . K . . . . � ν ν ν ( t ) = ν ( m k ) ν ν X n K m K E K k =1 • Wilson-Narayanan-Pfister-Sprintson ’10, Nazer-Gastpar ’11: Decoding is successful if the rates satisfy � 1 � R k < 1 2 log + 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 X n 1 m 1 E 1 Gaussian MACs? Z n X n Y n • Model using unequal m 2 2 E 2 ˆ D t power constraints: . . E � x ℓ � 2 ≤ nP ℓ . . . K . . � ν � ν � ν ν ( t ) = 0 ν ν ( m k ) X n m K K E K k =1

Computation over Gaussian MACs • How about general X n 1 m 1 E 1 Gaussian MACs? Z n X n Y n • Model using unequal m 2 2 E 2 ˆ D t power constraints: . . . . E � x ℓ � 2 ≤ nP ℓ . . . . . K . . . . � ν � ν � ν ν ( t ) = 0 ν ν ( m k ) X n m K K E K k =1 • 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 X n 1 m 1 E 1 Gaussian MACs? Z n X n Y n • Model using unequal m 2 2 E 2 ˆ D t power constraints: . . . . E � x ℓ � 2 ≤ nP ℓ . . . . . K . . . . � ν � ν � ν ν ( t ) = 0 ν ν ( m k ) X n m K K E K k =1 • 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 � � R ℓ < 1 P ℓ 2 log + + P ℓ . � L i =1 P i

Computation over Gaussian MACs • How about general X n 1 m 1 E 1 Gaussian MACs? Z n X n Y n • Model using unequal m 2 2 E 2 ˆ D t power constraints: . . . . E � x ℓ � 2 ≤ nP ℓ . . . . . K . . . . � ν � ν � ν ν ( t ) = 0 ν ν ( m k ) X n m K K E K k =1 • 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 � � R ℓ < 1 P ℓ 2 log + + P ℓ . � L i =1 P i • Nazer-Cadambe-Ntranos-Caire ’15: Expanded compute-and-forward framework to link unequal power setting to finite fields.

Point-to-Point Channels X n Y n p Y | X ˆ M Encoder Decoder M • Messages: m ∈ [2 nR ] � { 0 , . . . , 2 nR − 1 } • Encoder: a mapping x n ( m ) ∈ X n for each m ∈ [2 nR ] m ( y n ) ∈ [2 nR ] for each y n ∈ Y n • Decoder: a mapping ˆ

Point-to-Point Channels X n Y n p Y | X ˆ M Encoder Decoder M • Messages: m ∈ [2 nR ] � { 0 , . . . , 2 nR − 1 } • Encoder: a mapping x n ( m ) ∈ X n for each m ∈ [2 nR ] m ( y n ) ∈ [2 nR ] for each y n ∈ Y n • Decoder: a mapping ˆ Theorem (Shannon ’48) C = max p X ( x ) I ( X ; Y )

Point-to-Point Channels X n Y n p Y | X ˆ M Encoder Decoder M • Messages: m ∈ [2 nR ] � { 0 , . . . , 2 nR − 1 } • Encoder: a mapping x n ( m ) ∈ X n for each m ∈ [2 nR ] m ( y n ) ∈ [2 nR ] for each y n ∈ Y n • Decoder: a mapping ˆ Theorem (Shannon ’48) C = max p X ( x ) I ( X ; Y ) • Proof relies on random i.i.d. codebooks combined with joint typicality decoding.

Random i.i.d. Codebooks T ( n ) ( X ) ǫ X n Random i.i.d. Codes • Codewords are independent of one another. • Can directly target an input distribution p X ( x ) .

Point-to-Point Channels: Linear Codes U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder Code Construction:

Point-to-Point Channels: Linear Codes U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder Code Construction: • Pick a finite field F q and a symbol mapping x : F q → X .

Point-to-Point Channels: Linear Codes U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder Code Construction: • Pick a finite field F q and a symbol mapping x : F q → X . • Set κ = nR/ log( q ) .

Point-to-Point Channels: Linear Codes U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder Code Construction: • Pick a finite field F q and a symbol mapping x : F q → X . • Set κ = nR/ log( q ) . • Draw a random generator matrix G ∈ F κ × n elementwise q i.i.d. Unif( F q ) . Let G be a realization.

Point-to-Point Channels: Linear Codes U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder Code Construction: • Pick a finite field F q and a symbol mapping x : F q → X . • Set κ = nR/ log( q ) . • Draw a random generator matrix G ∈ F κ × n elementwise q i.i.d. Unif( F q ) . Let G be a realization. • Draw a random shift (or “dither”) D n elementwise i.i.d. Unif( F q ) . Let d n be a realization.

Point-to-Point Channels: Linear Codes U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder Code Construction: • Pick a finite field F q and a symbol mapping x : F q → X . • Set κ = nR/ log( q ) . • Draw a random generator matrix G ∈ F κ × n elementwise q i.i.d. Unif( F q ) . Let G be a realization. • Draw a random shift (or “dither”) D n elementwise i.i.d. Unif( F q ) . Let d n be a realization. ν ( m ) ∈ F κ • Take q-ary expansion of message m into the vector ν ν q .

Point-to-Point Channels: Linear Codes U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder Code Construction: • Pick a finite field F q and a symbol mapping x : F q → X . • Set κ = nR/ log( q ) . • Draw a random generator matrix G ∈ F κ × n elementwise q i.i.d. Unif( F q ) . Let G be a realization. • Draw a random shift (or “dither”) D n elementwise i.i.d. Unif( F q ) . Let d n be a realization. ν ( m ) ∈ F κ • Take q-ary expansion of message m into the vector ν ν q . • Linear codeword for message m is u n ( m ) = ν ν ( m ) G ⊕ d n . ν

Point-to-Point Channels: Linear Codes U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder Code Construction: • Pick a finite field F q and a symbol mapping x : F q → X . • Set κ = nR/ log( q ) . • Draw a random generator matrix G ∈ F κ × n elementwise q i.i.d. Unif( F q ) . Let G be a realization. • Draw a random shift (or “dither”) D n elementwise i.i.d. Unif( F q ) . Let d n be a realization. ν ( m ) ∈ F κ • Take q-ary expansion of message m into the vector ν ν q . • Linear codeword for message m is u n ( m ) = ν ν ( m ) G ⊕ d n . ν • Channel input at time i is x i ( m ) = x ( u i ( m )) .

Random i.i.d. Codebooks T ( n ) ( X ) ǫ X n Random Linear Codes • Codewords are pairwise independent of one another. • Codewords are uniformly distributed over F n q .

Point-to-Point Channels: Linear Codes U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder • 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 U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder • Well known that a direct application of linear coding is not sufficient to reach the point-to-point capacity, Ahlswede ’71. • Gallager ’68: Pick F q with q ≫ X and choose symbol mapping x ( u ) to reach c.a.i.d. from Unif( F q ) . This can attain the capacity.

Point-to-Point Channels: Linear Codes U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder • Well known that a direct application of linear coding is not sufficient to reach the point-to-point capacity, Ahlswede ’71. • Gallager ’68: Pick F q with q ≫ X and choose symbol mapping x ( u ) to reach c.a.i.d. from Unif( F q ) . 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 U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder • Well known that a direct application of linear coding is not sufficient to reach the point-to-point capacity, Ahlswede ’71. • Gallager ’68: Pick F q with q ≫ X and choose symbol mapping x ( u ) to reach c.a.i.d. from Unif( F q ) . 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 U n X n Y n Linear p Y | X ˆ x ( u ) M M Decoder Code Encoder • Well known that a direct application of linear coding is not sufficient to reach the point-to-point capacity, Ahlswede ’71. • Gallager ’68: Pick F q with q ≫ X and choose symbol mapping x ( u ) to reach c.a.i.d. from Unif( F q ) . 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 U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Code Construction:

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Code Construction: • Messages m ∈ [2 nR ] and auxiliary indices l ∈ [2 n ˆ R ] .

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Code Construction: • Messages m ∈ [2 nR ] and auxiliary indices l ∈ [2 n ˆ R ] . • Set κ = n ( R + ˆ R ) / log( q ) .

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Code Construction: • Messages m ∈ [2 nR ] and auxiliary indices l ∈ [2 n ˆ R ] . • Set κ = n ( R + ˆ R ) / log( q ) . • Pick generator matrix G and dither d n as before.

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Code Construction: • Messages m ∈ [2 nR ] and auxiliary indices l ∈ [2 n ˆ R ] . • Set κ = n ( R + ˆ R ) / log( q ) . • Pick generator matrix G and dither d n as before. ∈ F κ • Take q-ary expansions � ν ν � ν ν ( m ) ν ν ( l ) q .

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Code Construction: • Messages m ∈ [2 nR ] and auxiliary indices l ∈ [2 n ˆ R ] . • Set κ = n ( R + ˆ R ) / log( q ) . • Pick generator matrix G and dither d n as before. ∈ F κ • Take q-ary expansions � ν ν � ν ν ( m ) ν ν ( l ) q . • Linear codewords: u n ( m, l ) = � � G ⊕ d n . ν ν ν ( m ) ν ν ν ( l )

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Encoding:

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Encoding: • Fix p ( u ) and x ( u ) .

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Encoding: • Fix p ( u ) and x ( u ) . • Multicoding: For each m , find an index l such that u n ( m, l ) ∈ T ( n ) ( U ) ǫ ′

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Encoding: • Fix p ( u ) and x ( u ) . • Multicoding: For each m , find an index l such that u n ( m, l ) ∈ T ( n ) ( U ) ǫ ′ • Succeeds w.h.p. if ˆ R > D ( p U � p q ) (where p q is uniform over F q ).

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Encoding: • Fix p ( u ) and x ( u ) . • Multicoding: For each m , find an index l such that u n ( m, l ) ∈ T ( n ) ( U ) ǫ ′ • Succeeds w.h.p. if ˆ R > D ( p U � p q ) (where p q is uniform over F q ). • Transmit x i = x � � u i ( m, l ) .

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Encoding: • Fix p ( u ) and x ( u ) . • Multicoding: For each m , find an index l such that u n ( m, l ) ∈ T ( n ) ( U ) ǫ ′ • Succeeds w.h.p. if ˆ R > D ( p U � p q ) (where p q is uniform over F q ). • Transmit x i = x � � u i ( m, l ) . Decoding:

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Encoding: • Fix p ( u ) and x ( u ) . • Multicoding: For each m , find an index l such that u n ( m, l ) ∈ T ( n ) ( U ) ǫ ′ • Succeeds w.h.p. if ˆ R > D ( p U � p q ) (where p q is uniform over F q ). • Transmit x i = x � � u i ( m, l ) . Decoding: • Joint Typicality Decoding: Find the unique index ˆ m such that m, ˆ l ) , y n ) ∈ T ( n ) ( U, Y ) for some index ˆ � u n ( ˆ l . ǫ

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Encoding: • Fix p ( u ) and x ( u ) . • Multicoding: For each m , find an index l such that u n ( m, l ) ∈ T ( n ) ( U ) ǫ ′ • Succeeds w.h.p. if ˆ R > D ( p U � p q ) (where p q is uniform over F q ). • Transmit x i = x � � u i ( m, l ) . Decoding: • Joint Typicality Decoding: Find the unique index ˆ m such that m, ˆ l ) , y n ) ∈ T ( n ) ( U, Y ) for some index ˆ � u n ( ˆ l . ǫ • Succeeds w.h.p. if R + ˆ R < I ( U ; Y ) + D ( p U � p q )

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Theorem (Padakandla-Pradhan ’13) Any rate R satisfying R < p ( u ) , x ( u ) I ( U ; Y ) max is achievable. This is equal to the capacity if q ≥ |X| .

Point-to-Point Channels: Linear Codes + Multicoding U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Theorem (Padakandla-Pradhan ’13) Any rate R satisfying R < p ( u ) , x ( u ) I ( U ; Y ) max 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 U n X n Y n Linear Multi- p Y | X ˆ x ( u ) M Decoder M coding Code Encoder Theorem (Padakandla-Pradhan ’13) Any rate R satisfying R < p ( u ) , x ( u ) I ( U ; Y ) max 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 U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Code Construction: • Messages m k ∈ [2 nR k ] and auxiliary indices l k ∈ [2 n ˆ R k ] , k = 1 , 2 .

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Code Construction: • Messages m k ∈ [2 nR k ] and auxiliary indices l k ∈ [2 n ˆ R k ] , k = 1 , 2 . • Set κ = n (max { R 1 + ˆ R 1 , R 2 + ˆ R 2 } ) / log( q ) .

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Code Construction: • Messages m k ∈ [2 nR k ] and auxiliary indices l k ∈ [2 n ˆ R k ] , k = 1 , 2 . • Set κ = n (max { R 1 + ˆ R 1 , R 2 + ˆ R 2 } ) / log( q ) . • Pick generator matrix G and dithers d n 1 , d n 2 as before.

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Code Construction: • Messages m k ∈ [2 nR k ] and auxiliary indices l k ∈ [2 n ˆ R k ] , k = 1 , 2 . • Set κ = n (max { R 1 + ˆ R 1 , R 2 + ˆ R 2 } ) / log( q ) . • Pick generator matrix G and dithers d n 1 , d n 2 as before. ∈ F κ • Take q-ary expansions � � ν ν ν ( m 1 ) ν ν ( l 1 ) ν q ∈ F κ � ν ν � ν ν ( m 2 ) ν ν ( l 2 ) 0 Zero-padding q

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Code Construction: • Messages m k ∈ [2 nR k ] and auxiliary indices l k ∈ [2 n ˆ R k ] , k = 1 , 2 . • Set κ = n (max { R 1 + ˆ R 1 , R 2 + ˆ R 2 } ) / log( q ) . • Pick generator matrix G and dithers d n 1 , d n 2 as before. ∈ F κ • Take q-ary expansions � � η η η ( m 1 , l 1 ) q ∈ F κ � η � η η ( m 2 , l 2 ) q

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Code Construction: • Messages m k ∈ [2 nR k ] and auxiliary indices l k ∈ [2 n ˆ R k ] , k = 1 , 2 . • Set κ = n (max { R 1 + ˆ R 1 , R 2 + ˆ R 2 } ) / log( q ) . • Pick generator matrix G and dithers d n 1 , d n 2 as before. ∈ F κ • Take q-ary expansions � � η η η ( m 1 , l 1 ) q ∈ F κ � η � η η ( m 2 , l 2 ) q • Linear codewords: u n η ( m 1 , l 1 ) G ⊕ d n η 1 ( m 1 , l 1 ) = η 1 u n η ( m 2 , l 2 ) G ⊕ d n 2 ( m 2 , l 2 ) = η η 2

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Encoding:

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Encoding: • Fix p ( u 1 ) , p ( u 2 ) , x 1 ( u 1 ) , and x 2 ( u 2 ) .

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Encoding: • Fix p ( u 1 ) , p ( u 2 ) , x 1 ( u 1 ) , and x 2 ( u 2 ) . • Multicoding: For each m k , find an index l k such that k ( m k , l k ) ∈ T ( n ) u n ( U k ) . ǫ ′

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Encoding: • Fix p ( u 1 ) , p ( u 2 ) , x 1 ( u 1 ) , and x 2 ( u 2 ) . • Multicoding: For each m k , find an index l k such that k ( m k , l k ) ∈ T ( n ) u n ( U k ) . ǫ ′ • Succeeds w.h.p. if ˆ R k > D ( p U k � p q ) .

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Encoding: • Fix p ( u 1 ) , p ( u 2 ) , x 1 ( u 1 ) , and x 2 ( u 2 ) . • Multicoding: For each m k , find an index l k such that k ( m k , l k ) ∈ T ( n ) u n ( U k ) . ǫ ′ • Succeeds w.h.p. if ˆ R k > D ( p U k � p q ) . � � • Transmit x ki = x k u ki ( m k , l k ) .

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Encoding: • Fix p ( u 1 ) , p ( u 2 ) , x 1 ( u 1 ) , and x 2 ( u 2 ) . • Multicoding: For each m k , find an index l k such that k ( m k , l k ) ∈ T ( n ) u n ( U k ) . ǫ ′ • Succeeds w.h.p. if ˆ R k > D ( p U k � p q ) . � � • Transmit x ki = x k u ki ( m k , l k ) .

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Computation Problem:

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Computation Problem: • Consider the coefficients a ∈ F 2 q , a = [ a 1 , a 2 ]

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Computation Problem: • Consider the coefficients a ∈ F 2 q , a = [ a 1 , a 2 ] • For m k ∈ [2 nR k ] , l k ∈ [2 n ˆ R k ] , the linear combination of codewords with coefficient vector a is a 1 u n 1 ( m 1 , l 1 ) ⊕ a 2 u n 2 ( m 2 , l 2 ) G ⊕ a 1 d n 1 ⊕ a 2 d n � � = a 1 η η η ( m 1 , l 1 ) ⊕ a 2 η η ( m 2 , l 2 ) η 2 ν ( t ) G ⊕ d n = ν ν w t ∈ [2 n max { R 1 + ˆ R 1 ,R 2 + ˆ R 2 } ] = w n ( t ) ,

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Computation Problem: • Let M k be the chosen message and L k the chosen index from the multicoding step.

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Computation Problem: • Let M k be the chosen message and L k the chosen index from the multicoding step. • Decoder wants a linear combination of the codewords: W n ( T ) = a 1 U n 1 ( M 1 , L 1 ) ⊕ a 2 U n 2 ( M 2 , L 2 )

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Computation Problem: • Let M k be the chosen message and L k the chosen index from the multicoding step. • Decoder wants a linear combination of the codewords: W n ( T ) = a 1 U n 1 ( M 1 , L 1 ) ⊕ a 2 U n 2 ( M 2 , L 2 ) t ( y n ) ∈ [2 n max { R 1 + ˆ R 1 ,R 2 + ˆ R 2 } ] , y n ∈ Y n • Decoder: ˆ • Probability of Error: P ( n ) = P { T � = ˆ T } ǫ

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Computation Problem: • Let M k be the chosen message and L k the chosen index from the multicoding step. • Decoder wants a linear combination of the codewords: W n ( T ) = a 1 U n 1 ( M 1 , L 1 ) ⊕ a 2 U n 2 ( M 2 , L 2 ) t ( y n ) ∈ [2 n max { R 1 + ˆ R 1 ,R 2 + ˆ R 2 } ] , y n ∈ Y n • Decoder: ˆ • Probability of Error: P ( n ) = P { T � = ˆ T } ǫ • A rate pair is achievable if there exists a sequence of codes such that P ( n ) → 0 as n → ∞ . ǫ

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Decoding: • Joint Typicality Decoding: Find an index t ∈ [2 n max( R 1 + ˆ R 1 ,R 2 + ˆ R 2 ) ] such that ( w n ( t ) , y n ) ∈ T ( n ) . ǫ

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Theorem (Lim-Chen-Nazer-Gastpar Allerton ’15) A rate pair ( R 1 , R 2 ) is achievable if R 1 < I ( W ; Y ) − I ( W ; U 2 ) , R 2 < I ( W ; Y ) − I ( W ; U 1 ) , for some p ( u 1 ) p ( u 2 ) and functions x 1 ( u 1 ) , x 2 ( u 2 ) , where U k = F q , k = 1 , 2 , and W = a 1 U 1 ⊕ a 2 U 2 .

Nested Linear Coding Architecture U n X n Linear Multi- 1 1 M 1 x 1 ( u 1 ) coding Code p Y | X 1 X 2 Y n ˆ T Decoder U n Linear Multi- 2 M 2 x 2 ( u 2 ) coding Code X n 2 Theorem (Lim-Chen-Nazer-Gastpar Allerton ’15) A rate pair ( R 1 , R 2 ) is achievable if R 1 < I ( W ; Y ) − I ( W ; U 2 ) , R 2 < I ( W ; Y ) − I ( W ; U 1 ) , for some p ( u 1 ) p ( u 2 ) and functions x 1 ( u 1 ) , x 2 ( u 2 ) , where U k = F q , k = 1 , 2 , and W = a 1 U 1 ⊕ a 2 U 2 . • Padakandla-Pradhan ’13: Special case where R 1 = R 2 .

Proof Sketch • WLOG assume M = { M 1 = 0 , M 2 = 0 , L 1 = 0 , L 2 = 0 } .

Proof Sketch • WLOG assume M = { M 1 = 0 , M 2 = 0 , L 1 = 0 , L 2 = 0 } . • Union bound: P ( n ) � ( W n ( t ) , Y n ) ∈ T ( n ) � � ≤ P |M . ǫ ǫ t � =0

Proof Sketch • WLOG assume M = { M 1 = 0 , M 2 = 0 , L 1 = 0 , L 2 = 0 } . • Union bound: P ( n ) � ( W n ( t ) , Y n ) ∈ T ( n ) � � ≤ P |M . ǫ ǫ t � =0 • Notice that the L k depend on the codebook so Y n and W n ( t ) are not independent.

Proof Sketch • WLOG assume M = { M 1 = 0 , M 2 = 0 , L 1 = 0 , L 2 = 0 } . • Union bound: P ( n ) � ( W n ( t ) , Y n ) ∈ T ( n ) � � ≤ P |M . ǫ ǫ t � =0 • Notice that the L k depend on the codebook so Y n and W n ( t ) are not independent. • To get around this issue, we analyze � ( W n ( t ) , Y n ) ∈ T ( n ) , U n 1 (0 , 0) ∈ T ( n ) , U n 2 (0 , 0) ∈ T ( n ) P ( E ) = � |M � P ǫ ǫ ǫ t � =0

Proof Sketch • WLOG assume M = { M 1 = 0 , M 2 = 0 , L 1 = 0 , L 2 = 0 } . • Union bound: P ( n ) � ( W n ( t ) , Y n ) ∈ T ( n ) � � ≤ P |M . ǫ ǫ t � =0 • Notice that the L k depend on the codebook so Y n and W n ( t ) are not independent. • To get around this issue, we analyze � ( W n ( t ) , Y n ) ∈ T ( n ) , U n 1 (0 , 0) ∈ T ( n ) , U n 2 (0 , 0) ∈ T ( n ) P ( E ) = � |M � P ǫ ǫ ǫ t � =0 • Conditioned on M , Y n → ( U n 1 (0 , 0) , U n 2 (0 , 0)) → W n ( t )

Proof Sketch • WLOG assume M = { M 1 = 0 , M 2 = 0 , L 1 = 0 , L 2 = 0 } . • Union bound: P ( n ) � ( W n ( t ) , Y n ) ∈ T ( n ) � � ≤ P |M . ǫ ǫ t � =0 • Notice that the L k depend on the codebook so Y n and W n ( t ) are not independent. • To get around this issue, we analyze � ( W n ( t ) , Y n ) ∈ T ( n ) , U n 1 (0 , 0) ∈ T ( n ) , U n 2 (0 , 0) ∈ T ( n ) P ( E ) = � |M � P ǫ ǫ ǫ t � =0 • 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 R k + ˆ R k + ˆ R 1 + ˆ R 2 < I ( W ; Y ) + D ( p W || p q ) + D ( p U 1 || p q ) + D ( p U 2 || p q )

Compute-and-Forward over a Gaussian MAC • Consider a Gaussian MAC with real-valued channel output Y = h 1 X 1 + h 2 X 2 + Z

Compute-and-Forward over a Gaussian MAC • Consider a Gaussian MAC with real-valued channel output Y = h 1 X 1 + h 2 X 2 + Z • Want to recover a 1 X n 1 + a 2 X n 2 for some integers a 1 , a 2 .