Pliable Index Coding - nicest.lab.uic.edu€¦ · Tang Liu, PhD thesis defense, Nov 19th 2019, UIC Broadcast channel with message side information • Broadcast nature of wireless

Tang Liu, PhD thesis defense, Nov 19th 2019, UIC

Pliable Index Coding

Tang Liu University of Illinois at Chicago

11/19/2019

1

• Introduction• Index Coding problem• Pliable Index Coding problem• Related past work and summary of contributions

• Contributions• Greedy achievability for a constant fraction of satisfied users• Complete-S and circular-arc PICOD• Decentralized PICOD• Individual-message secure PICOD

• Conclusion/Discussion/Future work

Content

2 Tang Liu, PhD thesis defense, Nov 19th 2019, UIC


Motivation

3


Broadcast channel with message side information

• Broadcast nature of wireless communication.• Wireless communication is a crucial part of wireless communication network.• Broadcast Internet services (e.g. video/audio streaming) are popular.

• Receivers have more powerful processor and more storage.• Devices (e.g. smartphone) become more and more powerful.• Storage becomes cheaper and cheaper.

The coding scheme at the transmitter can leverage the computation and storage resources and the receiver.

4


Satellite communication

Figure: Satellite communication network with basestations from different

[1] Y. Birk and T. Kol. Coding-on-demand by an informed source (ISCOD) for efficient broadcast of different supplemental data to caching clients. IEEE Transactions on Information Theory, 52(6):2825–2830, 2006. Earlier version appeared in INFOCOM ’98.

5


Index coding

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}, 𝒟 = {D1, …, Dn}

…WA1

WA2WAnD1 D2 Dn

Notation: ;

;Side information set: Fixed desired message set: .

[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];Di ⊆ [m]∖Ai

6


xℓk = ENC(W, 𝒜, 𝒟)

Index coding

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}, 𝒟 = {D1, …, Dn}

…WA1

WA2WAnD1 D2 Dn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}


6



Index coding

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}, 𝒟 = {D1, …, Dn}

…WA1

WA2WAnD1 D2 Dn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}


6


Index coding

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}, 𝒟 = {D1, …, Dn}

…WA1

WA2WAn

wD1= DEC1(WA1

, xℓk)

D1 D2 Dn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}


6



Index coding

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}, 𝒟 = {D1, …, Dn}

…WA1

WA2WAn

wD1= DEC1(WA1

, xℓk)

D1 D2 Dn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}


6


Index coding

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}, 𝒟 = {D1, …, Dn}

…WA1

WA2WAn

wD1= DEC1(WA1

, xℓk) wD2= DEC2(WA2

, xℓk)

D1 D2 Dn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}


6



Index coding

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}, 𝒟 = {D1, …, Dn}

…WA1

WA2WAn

wD1= DEC1(WA1


, xℓk)

D1 D2 Dn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}


6


Index coding

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}, 𝒟 = {D1, …, Dn}

…WA1

WA2WAn

wD1= DEC1(WA1


, xℓk) wDn= DECn(WAn

, xℓk)

D1 D2 Dn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}


6


Index coding

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}, 𝒟 = {D1, …, Dn}

…WA1

WA2WAn

wD1= DEC1(WA1



, xℓk)

D1 D2 Dn

Goal: find the minimum number of transmission needed by the transmitter such that all users can decode their desired messages

ℓ*

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}


6


Pliability in the choice of desired message

• There are scenarios where users do not want a specific message.

• The transmitter can choose the desired message for the users to reduce the number of transmissions, as long as users do not have the messages already.

• Goal: find the one with the shortest codelength among all index coding instances with the same users’ side information sets.

7


Application: advertising network

• All advertisements are distributed by a single transmitter.

• Consumers may have preference, but do not request a specific ad.

• Distribution/transmission of the ads can leverage this pliability.

Figure: Advertising network.

8


Pliable Index CODing (PICOD(t))

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

Notation: ;

;Side information set: Pliable desired message set:some s.t.

[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

9


xℓk = ENC(W, 𝒜)


User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

9




User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

9



User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1

, xℓk)

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

9




User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1

, xℓk)

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

9



User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1


, xℓk)

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

9




User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1


, xℓk)

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

9



User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1



, xℓk)

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

9



User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1



, xℓk)

Goal: find the minimum number of transmission needed by the transmitter such that each user can decode messages outside its side information set.

ℓ*t

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

9


Decentralized system

• No centralized controller/distributer/transmitter. All transmissions are done by the users in the system.

• The codewords are generated by the users based on their side information set. Decentralized system is “weaker” compared to centralized system in terms of transmission rate.

• Examples: ad-hoc network, p2p network, distributed storage…

10


Application of decentralized system

Figure: Distributed music by Spotify11


Decentralized PICOD(t)

User 2User 1 User n

W = {w1, …, wm}, 𝒜 = {A1, …, An}, xℓk = {xℓ1k, …, xℓnk}

…WA1

WA2WAn

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜


xℓ1k = ENC1(WA1, 𝒜)


User 2User 1 User n


…WA1

WA2WAn

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜




User 2User 1 User n


…WA1

WA2WAn

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜



User 2User 1 User n


…WA1

WA2WAn

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜




User 2User 1 User n


…WA1

WA2WAn

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜




User 2User 1 User n


…WA1

WA2WAn

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜



User 2User 1 User n


…WA1

WA2WAn

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜


xℓnk = ENCn(WAn, 𝒜)


User 2User 1 User n


…WA1

WA2WAn

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜


xℓnk = ENCn(WAn, 𝒜)


User 2User 1 User n


…WA1

WA2WAn

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜



User 2User 1 User n


…WA1

WA2WAn

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜



User 2User 1 User n


…WA1

WA2WAn

wD1= DEC1(WA1

, xℓk)

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜



User 2User 1 User n


…WA1

WA2WAn

wD1= DEC1(WA1


, xℓk)

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜



User 2User 1 User n


…WA1

WA2WAn

wD1= DEC1(WA1



, xℓk)

12

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜



User 2User 1 User n


…WA1

WA2WAn

wD1= DEC1(WA1



, xℓk)

12

• The codewords are generated by the users based on their side information sets and globally known side information structure.

• Goal: determine the minimum number of transmissions that allow each user to decode messages outside its side information.t

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

𝒜 𝒜 𝒜


Security in PICOD• Security is a major concern in today’s communication. A message should not be

decoded at the users for which the message is not intended for.

• Information theoretical security provides guarantee regardless of the computational power of the attacker/eavesdropper/non-intended party.

• In broadcast channels, without explicitly imposing security constraints, the information is likely to be overheard by multiple users.

• Strong security constraint:

• Individual-message security constraint:

H(W[m]∖(Ai∪Di) |xℓk, WAi) ≥ H(W[m]∖(Ai∪Di)) − ϵk, ∀i ∈ [n]

H(wj |xℓk, WAi) ≥ H(wj) − ϵk, ∀i ∈ [n], ∀j ∈ [m]∖(Ai ∪ Di)

13


Secure PICOD(t)

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

14



Secure PICOD(t)

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

14



Secure PICOD(t)

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

14


Secure PICOD(t)

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1

, xℓk)

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

14



Secure PICOD(t)

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1

, xℓk)

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

14


Secure PICOD(t)

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1


, xℓk)

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

14



Secure PICOD(t)

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1


, xℓk)

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

14


Secure PICOD(t)

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1



, xℓk)

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

14


Secure PICOD(t)

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1



, xℓk)

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

14

I(wj; xℓk, WAi) ≤ ϵk, ∀i ∈ [n], ∀j ∈ [m]∖(Ai ∪ {di})


Secure PICOD(t)

User 2User 1 User n

Transmitter

W = {w1, …, wm}, 𝒜 = {A1, …, An}

…WA1

WA2WAn

wD1= DEC1(WA1



, xℓk)

Goal: find the minimum number of transmission needed by the transmitter such that each user can decode messages outside its side information and no more than .

ℓ*t

t

Notation: ;


[m] := {1,2,…, m}w ∈ 𝔽2k, WA = {wi, i ∈ A}

Ai ⊂ [m];

Di ⊆ [m]∖Ai |Di | = t .

14

I(wj; xℓk, WAi) ≤ ϵk, ∀i ∈ [n], ∀j ∈ [m]∖(Ai ∪ {di})


Tx

User 1 User 2 User 3 User 4

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A2 = {2}<latexit sha1_base64="Utx/LcoVW2jynbZvBeT45nrG7j4=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQadc+tew8XteZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHnrmQxQ==</latexit><latexit sha1_base64="Utx/LcoVW2jynbZvBeT45nrG7j4=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQadc+tew8XteZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHnrmQxQ==</latexit><latexit sha1_base64="Utx/LcoVW2jynbZvBeT45nrG7j4=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQadc+tew8XteZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHnrmQxQ==</latexit><latexit sha1_base64="Utx/LcoVW2jynbZvBeT45nrG7j4=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQadc+tew8XteZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHnrmQxQ==</latexit>

A3 = {3}<latexit sha1_base64="NdY91Z7nBx1SnRS8estTbjzY/GQ=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU1ItQ8eKxgv2AJJTNdtsu3eyG3Y1QQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZFyWcaeO6305pZXVtfaO8Wdna3tndq+4ftLVMFaEtIrlU3QhrypmgLcMMp91EURxHnHai8d3U7zxRpZkUj2aS0DDGQ8EGjGBjJT+77Z3nN0F2HuS9as2tuzOgZeIVpAYFmr3qV9CXJI2pMIRjrX3PTUyYYWUY4TSvBKmmCSZjPKS+pQLHVIfZ7OQcnViljwZS2RIGzdTfExmOtZ7Eke2MsRnpRW8q/uf5qRlchxkTSWqoIPNFg5QjI9H0f9RnihLDJ5Zgopi9FZERVpgYm1LFhuAtvrxM2md1z617Dxe1xmURRxmO4BhOwYMraMA9NKEFBCQ8wyu8OcZ5cd6dj3lrySlmDuEPnM8focqQxw==</latexit><latexit sha1_base64="NdY91Z7nBx1SnRS8estTbjzY/GQ=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU1ItQ8eKxgv2AJJTNdtsu3eyG3Y1QQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZFyWcaeO6305pZXVtfaO8Wdna3tndq+4ftLVMFaEtIrlU3QhrypmgLcMMp91EURxHnHai8d3U7zxRpZkUj2aS0DDGQ8EGjGBjJT+77Z3nN0F2HuS9as2tuzOgZeIVpAYFmr3qV9CXJI2pMIRjrX3PTUyYYWUY4TSvBKmmCSZjPKS+pQLHVIfZ7OQcnViljwZS2RIGzdTfExmOtZ7Eke2MsRnpRW8q/uf5qRlchxkTSWqoIPNFg5QjI9H0f9RnihLDJ5Zgopi9FZERVpgYm1LFhuAtvrxM2md1z617Dxe1xmURRxmO4BhOwYMraMA9NKEFBCQ8wyu8OcZ5cd6dj3lrySlmDuEPnM8focqQxw==</latexit><latexit sha1_base64="NdY91Z7nBx1SnRS8estTbjzY/GQ=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU1ItQ8eKxgv2AJJTNdtsu3eyG3Y1QQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZFyWcaeO6305pZXVtfaO8Wdna3tndq+4ftLVMFaEtIrlU3QhrypmgLcMMp91EURxHnHai8d3U7zxRpZkUj2aS0DDGQ8EGjGBjJT+77Z3nN0F2HuS9as2tuzOgZeIVpAYFmr3qV9CXJI2pMIRjrX3PTUyYYWUY4TSvBKmmCSZjPKS+pQLHVIfZ7OQcnViljwZS2RIGzdTfExmOtZ7Eke2MsRnpRW8q/uf5qRlchxkTSWqoIPNFg5QjI9H0f9RnihLDJ5Zgopi9FZERVpgYm1LFhuAtvrxM2md1z617Dxe1xmURRxmO4BhOwYMraMA9NKEFBCQ8wyu8OcZ5cd6dj3lrySlmDuEPnM8focqQxw==</latexit><latexit sha1_base64="NdY91Z7nBx1SnRS8estTbjzY/GQ=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU1ItQ8eKxgv2AJJTNdtsu3eyG3Y1QQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZFyWcaeO6305pZXVtfaO8Wdna3tndq+4ftLVMFaEtIrlU3QhrypmgLcMMp91EURxHnHai8d3U7zxRpZkUj2aS0DDGQ8EGjGBjJT+77Z3nN0F2HuS9as2tuzOgZeIVpAYFmr3qV9CXJI2pMIRjrX3PTUyYYWUY4TSvBKmmCSZjPKS+pQLHVIfZ7OQcnViljwZS2RIGzdTfExmOtZ7Eke2MsRnpRW8q/uf5qRlchxkTSWqoIPNFg5QjI9H0f9RnihLDJ5Zgopi9FZERVpgYm1LFhuAtvrxM2md1z617Dxe1xmURRxmO4BhOwYMraMA9NKEFBCQ8wyu8OcZ5cd6dj3lrySlmDuEPnM8focqQxw==</latexit>

A4 = {4}<latexit sha1_base64="XOSD6CexLkbM9tv+kGEbHWgum2M=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mkqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQu6p5b9x4ateZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHpNuQyQ==</latexit><latexit sha1_base64="XOSD6CexLkbM9tv+kGEbHWgum2M=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mkqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQu6p5b9x4ateZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHpNuQyQ==</latexit><latexit sha1_base64="XOSD6CexLkbM9tv+kGEbHWgum2M=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mkqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQu6p5b9x4ateZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHpNuQyQ==</latexit><latexit sha1_base64="XOSD6CexLkbM9tv+kGEbHWgum2M=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mkqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQu6p5b9x4ateZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHpNuQyQ==</latexit>

d4 = 1<latexit sha1_base64="mXFggRNWuGQ0rQO0bQBdOWsLq1s=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1ItQ8OKxgmkLbSibzaZdutmE3YlQSn+DFw+KePUHefPfuG1z0NYHA4/3ZpiZF2ZSGHTdb6e0tr6xuVXeruzs7u0fVA+PWibNNeM+S2WqOyE1XArFfRQoeSfTnCah5O1wdDfz209cG5GqRxxnPEjoQIlYMIpW8qP+5a3Xr9bcujsHWSVeQWpQoNmvfvWilOUJV8gkNabruRkGE6pRMMmnlV5ueEbZiA5411JFE26CyfzYKTmzSkTiVNtSSObq74kJTYwZJ6HtTCgOzbI3E//zujnGN8FEqCxHrthiUZxLgimZfU4ioTlDObaEMi3srYQNqaYMbT4VG4K3/PIqaV3UPbfuPVzWGldFHGU4gVM4Bw+uoQH30AQfGAh4hld4c5Tz4rw7H4vWklPMHMMfOJ8/4N2OBQ==</latexit><latexit sha1_base64="mXFggRNWuGQ0rQO0bQBdOWsLq1s=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1ItQ8OKxgmkLbSibzaZdutmE3YlQSn+DFw+KePUHefPfuG1z0NYHA4/3ZpiZF2ZSGHTdb6e0tr6xuVXeruzs7u0fVA+PWibNNeM+S2WqOyE1XArFfRQoeSfTnCah5O1wdDfz209cG5GqRxxnPEjoQIlYMIpW8qP+5a3Xr9bcujsHWSVeQWpQoNmvfvWilOUJV8gkNabruRkGE6pRMMmnlV5ueEbZiA5411JFE26CyfzYKTmzSkTiVNtSSObq74kJTYwZJ6HtTCgOzbI3E//zujnGN8FEqCxHrthiUZxLgimZfU4ioTlDObaEMi3srYQNqaYMbT4VG4K3/PIqaV3UPbfuPVzWGldFHGU4gVM4Bw+uoQH30AQfGAh4hld4c5Tz4rw7H4vWklPMHMMfOJ8/4N2OBQ==</latexit><latexit sha1_base64="mXFggRNWuGQ0rQO0bQBdOWsLq1s=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1ItQ8OKxgmkLbSibzaZdutmE3YlQSn+DFw+KePUHefPfuG1z0NYHA4/3ZpiZF2ZSGHTdb6e0tr6xuVXeruzs7u0fVA+PWibNNeM+S2WqOyE1XArFfRQoeSfTnCah5O1wdDfz209cG5GqRxxnPEjoQIlYMIpW8qP+5a3Xr9bcujsHWSVeQWpQoNmvfvWilOUJV8gkNabruRkGE6pRMMmnlV5ueEbZiA5411JFE26CyfzYKTmzSkTiVNtSSObq74kJTYwZJ6HtTCgOzbI3E//zujnGN8FEqCxHrthiUZxLgimZfU4ioTlDObaEMi3srYQNqaYMbT4VG4K3/PIqaV3UPbfuPVzWGldFHGU4gVM4Bw+uoQH30AQfGAh4hld4c5Tz4rw7H4vWklPMHMMfOJ8/4N2OBQ==</latexit><latexit sha1_base64="mXFggRNWuGQ0rQO0bQBdOWsLq1s=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1ItQ8OKxgmkLbSibzaZdutmE3YlQSn+DFw+KePUHefPfuG1z0NYHA4/3ZpiZF2ZSGHTdb6e0tr6xuVXeruzs7u0fVA+PWibNNeM+S2WqOyE1XArFfRQoeSfTnCah5O1wdDfz209cG5GqRxxnPEjoQIlYMIpW8qP+5a3Xr9bcujsHWSVeQWpQoNmvfvWilOUJV8gkNabruRkGE6pRMMmnlV5ueEbZiA5411JFE26CyfzYKTmzSkTiVNtSSObq74kJTYwZJ6HtTCgOzbI3E//zujnGN8FEqCxHrthiUZxLgimZfU4ioTlDObaEMi3srYQNqaYMbT4VG4K3/PIqaV3UPbfuPVzWGldFHGU4gVM4Bw+uoQH30AQfGAh4hld4c5Tz4rw7H4vWklPMHMMfOJ8/4N2OBQ==</latexit>

d1 = 2<latexit sha1_base64="W6UvBpEI7GtHd5wGf5yXIk4+fio=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9SIUvHisYNpCG8pms2mXbjZhdyKU0t/gxYMiXv1B3vw3btsctPXBwOO9GWbmhZkUBl3321lb39jc2i7tlHf39g8OK0fHLZPmmnGfpTLVnZAaLoXiPgqUvJNpTpNQ8nY4upv57SeujUjVI44zHiR0oEQsGEUr+VHfu633K1W35s5BVolXkCoUaPYrX70oZXnCFTJJjel6bobBhGoUTPJpuZcbnlE2ogPetVTRhJtgMj92Ss6tEpE41bYUkrn6e2JCE2PGSWg7E4pDs+zNxP+8bo7xTTARKsuRK7ZYFOeSYEpmn5NIaM5Qji2hTAt7K2FDqilDm0/ZhuAtv7xKWvWa59a8h8tq46qIowSncAYX4ME1NOAemuADAwHP8ApvjnJenHfnY9G65hQzJ/AHzucP3c+OAw==</latexit><latexit sha1_base64="W6UvBpEI7GtHd5wGf5yXIk4+fio=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9SIUvHisYNpCG8pms2mXbjZhdyKU0t/gxYMiXv1B3vw3btsctPXBwOO9GWbmhZkUBl3321lb39jc2i7tlHf39g8OK0fHLZPmmnGfpTLVnZAaLoXiPgqUvJNpTpNQ8nY4upv57SeujUjVI44zHiR0oEQsGEUr+VHfu633K1W35s5BVolXkCoUaPYrX70oZXnCFTJJjel6bobBhGoUTPJpuZcbnlE2ogPetVTRhJtgMj92Ss6tEpE41bYUkrn6e2JCE2PGSWg7E4pDs+zNxP+8bo7xTTARKsuRK7ZYFOeSYEpmn5NIaM5Qji2hTAt7K2FDqilDm0/ZhuAtv7xKWvWa59a8h8tq46qIowSncAYX4ME1NOAemuADAwHP8ApvjnJenHfnY9G65hQzJ/AHzucP3c+OAw==</latexit><latexit sha1_base64="W6UvBpEI7GtHd5wGf5yXIk4+fio=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9SIUvHisYNpCG8pms2mXbjZhdyKU0t/gxYMiXv1B3vw3btsctPXBwOO9GWbmhZkUBl3321lb39jc2i7tlHf39g8OK0fHLZPmmnGfpTLVnZAaLoXiPgqUvJNpTpNQ8nY4upv57SeujUjVI44zHiR0oEQsGEUr+VHfu633K1W35s5BVolXkCoUaPYrX70oZXnCFTJJjel6bobBhGoUTPJpuZcbnlE2ogPetVTRhJtgMj92Ss6tEpE41bYUkrn6e2JCE2PGSWg7E4pDs+zNxP+8bo7xTTARKsuRK7ZYFOeSYEpmn5NIaM5Qji2hTAt7K2FDqilDm0/ZhuAtv7xKWvWa59a8h8tq46qIowSncAYX4ME1NOAemuADAwHP8ApvjnJenHfnY9G65hQzJ/AHzucP3c+OAw==</latexit><latexit sha1_base64="W6UvBpEI7GtHd5wGf5yXIk4+fio=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9SIUvHisYNpCG8pms2mXbjZhdyKU0t/gxYMiXv1B3vw3btsctPXBwOO9GWbmhZkUBl3321lb39jc2i7tlHf39g8OK0fHLZPmmnGfpTLVnZAaLoXiPgqUvJNpTpNQ8nY4upv57SeujUjVI44zHiR0oEQsGEUr+VHfu633K1W35s5BVolXkCoUaPYrX70oZXnCFTJJjel6bobBhGoUTPJpuZcbnlE2ogPetVTRhJtgMj92Ss6tEpE41bYUkrn6e2JCE2PGSWg7E4pDs+zNxP+8bo7xTTARKsuRK7ZYFOeSYEpmn5NIaM5Qji2hTAt7K2FDqilDm0/ZhuAtv7xKWvWa59a8h8tq46qIowSncAYX4ME1NOAemuADAwHP8ApvjnJenHfnY9G65hQzJ/AHzucP3c+OAw==</latexit>

d2 = 3<latexit sha1_base64="cWjunNJMMdmF+rQUPTu7EjV0a80=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqqBeh4MVjBdMW2lA2m027dLMbdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NYyU4T6RHKpuiHWlDNBfcMMp91UUZyEnHbC8d3M7zxRpZkUj2aS0iDBQ8FiRrCxkh8NGrcXg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY617npuaIMfKMMLptNLPNE0xGeMh7VkqcEJ1kM+PnaIzq0QolsqWMGiu/p7IcaL1JAltZ4LNSC97M/E/r5eZ+CbImUgzQwVZLIozjoxEs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQbdc+tew+XteZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzhvDjvzseiteQUM8fwB87nD+DZjgU=</latexit><latexit sha1_base64="cWjunNJMMdmF+rQUPTu7EjV0a80=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqqBeh4MVjBdMW2lA2m027dLMbdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NYyU4T6RHKpuiHWlDNBfcMMp91UUZyEnHbC8d3M7zxRpZkUj2aS0iDBQ8FiRrCxkh8NGrcXg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY617npuaIMfKMMLptNLPNE0xGeMh7VkqcEJ1kM+PnaIzq0QolsqWMGiu/p7IcaL1JAltZ4LNSC97M/E/r5eZ+CbImUgzQwVZLIozjoxEs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQbdc+tew+XteZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzhvDjvzseiteQUM8fwB87nD+DZjgU=</latexit><latexit sha1_base64="cWjunNJMMdmF+rQUPTu7EjV0a80=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqqBeh4MVjBdMW2lA2m027dLMbdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NYyU4T6RHKpuiHWlDNBfcMMp91UUZyEnHbC8d3M7zxRpZkUj2aS0iDBQ8FiRrCxkh8NGrcXg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY617npuaIMfKMMLptNLPNE0xGeMh7VkqcEJ1kM+PnaIzq0QolsqWMGiu/p7IcaL1JAltZ4LNSC97M/E/r5eZ+CbImUgzQwVZLIozjoxEs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQbdc+tew+XteZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzhvDjvzseiteQUM8fwB87nD+DZjgU=</latexit><latexit sha1_base64="cWjunNJMMdmF+rQUPTu7EjV0a80=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqqBeh4MVjBdMW2lA2m027dLMbdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NYyU4T6RHKpuiHWlDNBfcMMp91UUZyEnHbC8d3M7zxRpZkUj2aS0iDBQ8FiRrCxkh8NGrcXg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY617npuaIMfKMMLptNLPNE0xGeMh7VkqcEJ1kM+PnaIzq0QolsqWMGiu/p7IcaL1JAltZ4LNSC97M/E/r5eZ+CbImUgzQwVZLIozjoxEs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQbdc+tew+XteZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzhvDjvzseiteQUM8fwB87nD+DZjgU=</latexit>

d3 = 4<latexit sha1_base64="Dzvoa4vpObiWJLCDzizaOBHfcxw=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0qBeh4MVjBdMW2lA2m027dHcTdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NZJpgj1ScIT1Q2xppxJ6htmOO2mimIRctoJx3czv/NElWaJfDSTlAYCDyWLGcHGSn40uLxtDKo1t+7OgVaJV5AaFGgNql/9KCGZoNIQjrXueW5qghwrwwin00o/0zTFZIyHtGepxILqIJ8fO0VnVolQnChb0qC5+nsix0LriQhtp8BmpJe9mfif18tMfBPkTKaZoZIsFsUZRyZBs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQv6p5b9x4ateZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzpvDjvzseiteQUM8fwB87nD+Pjjgc=</latexit><latexit sha1_base64="Dzvoa4vpObiWJLCDzizaOBHfcxw=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0qBeh4MVjBdMW2lA2m027dHcTdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NZJpgj1ScIT1Q2xppxJ6htmOO2mimIRctoJx3czv/NElWaJfDSTlAYCDyWLGcHGSn40uLxtDKo1t+7OgVaJV5AaFGgNql/9KCGZoNIQjrXueW5qghwrwwin00o/0zTFZIyHtGepxILqIJ8fO0VnVolQnChb0qC5+nsix0LriQhtp8BmpJe9mfif18tMfBPkTKaZoZIsFsUZRyZBs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQv6p5b9x4ateZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzpvDjvzseiteQUM8fwB87nD+Pjjgc=</latexit><latexit sha1_base64="Dzvoa4vpObiWJLCDzizaOBHfcxw=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0qBeh4MVjBdMW2lA2m027dHcTdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NZJpgj1ScIT1Q2xppxJ6htmOO2mimIRctoJx3czv/NElWaJfDSTlAYCDyWLGcHGSn40uLxtDKo1t+7OgVaJV5AaFGgNql/9KCGZoNIQjrXueW5qghwrwwin00o/0zTFZIyHtGepxILqIJ8fO0VnVolQnChb0qC5+nsix0LriQhtp8BmpJe9mfif18tMfBPkTKaZoZIsFsUZRyZBs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQv6p5b9x4ateZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzpvDjvzseiteQUM8fwB87nD+Pjjgc=</latexit><latexit sha1_base64="Dzvoa4vpObiWJLCDzizaOBHfcxw=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0qBeh4MVjBdMW2lA2m027dHcTdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NZJpgj1ScIT1Q2xppxJ6htmOO2mimIRctoJx3czv/NElWaJfDSTlAYCDyWLGcHGSn40uLxtDKo1t+7OgVaJV5AaFGgNql/9KCGZoNIQjrXueW5qghwrwwin00o/0zTFZIyHtGepxILqIJ8fO0VnVolQnChb0qC5+nsix0LriQhtp8BmpJe9mfif18tMfBPkTKaZoZIsFsUZRyZBs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQv6p5b9x4ateZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzpvDjvzseiteQUM8fwB87nD+Pjjgc=</latexit>

Figure: One optimal code is .{w1 + w2, w2 + w3, w3 + w4}

Index coding

15


Tx



A1 = {1}<latexit sha1_base64="L0mRe+qwbGYO6dvE8BizKDiIh0I=">AAAB8nicbVBNS8NAEN3Ur1q/qh69BIvgqWRF1ItQ8eKxgrWFJJTNdtMu3eyG3YlQQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZF6WCG/C8b6eysrq2vlHdrG1t7+zu1fcPHo3KNGUdqoTSvYgYJrhkHeAgWC/VjCSRYN1ofDv1u09MG67kA0xSFiZkKHnMKQEr+flNHxfXQY6Dol9veE1vBneZ4JI0UIl2v/4VDBTNEiaBCmKMj70Uwpxo4FSwohZkhqWEjsmQ+ZZKkjAT5rOTC/fEKgM3VtqWBHem/p7ISWLMJIlsZ0JgZBa9qfif52cQX4U5l2kGTNL5ojgTLih3+r874JpREBNLCNXc3urSEdGEgk2pZkPAiy8vk8ezJvaa+P680boo46iiI3SMThFGl6iF7lAbdRBFCj2jV/TmgPPivDsf89aKU84coj9wPn8Am6iQww==</latexit><latexit sha1_base64="L0mRe+qwbGYO6dvE8BizKDiIh0I=">AAAB8nicbVBNS8NAEN3Ur1q/qh69BIvgqWRF1ItQ8eKxgrWFJJTNdtMu3eyG3YlQQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZF6WCG/C8b6eysrq2vlHdrG1t7+zu1fcPHo3KNGUdqoTSvYgYJrhkHeAgWC/VjCSRYN1ofDv1u09MG67kA0xSFiZkKHnMKQEr+flNHxfXQY6Dol9veE1vBneZ4JI0UIl2v/4VDBTNEiaBCmKMj70Uwpxo4FSwohZkhqWEjsmQ+ZZKkjAT5rOTC/fEKgM3VtqWBHem/p7ISWLMJIlsZ0JgZBa9qfif52cQX4U5l2kGTNL5ojgTLih3+r874JpREBNLCNXc3urSEdGEgk2pZkPAiy8vk8ezJvaa+P680boo46iiI3SMThFGl6iF7lAbdRBFCj2jV/TmgPPivDsf89aKU84coj9wPn8Am6iQww==</latexit><latexit sha1_base64="L0mRe+qwbGYO6dvE8BizKDiIh0I=">AAAB8nicbVBNS8NAEN3Ur1q/qh69BIvgqWRF1ItQ8eKxgrWFJJTNdtMu3eyG3YlQQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZF6WCG/C8b6eysrq2vlHdrG1t7+zu1fcPHo3KNGUdqoTSvYgYJrhkHeAgWC/VjCSRYN1ofDv1u09MG67kA0xSFiZkKHnMKQEr+flNHxfXQY6Dol9veE1vBneZ4JI0UIl2v/4VDBTNEiaBCmKMj70Uwpxo4FSwohZkhqWEjsmQ+ZZKkjAT5rOTC/fEKgM3VtqWBHem/p7ISWLMJIlsZ0JgZBa9qfif52cQX4U5l2kGTNL5ojgTLih3+r874JpREBNLCNXc3urSEdGEgk2pZkPAiy8vk8ezJvaa+P680boo46iiI3SMThFGl6iF7lAbdRBFCj2jV/TmgPPivDsf89aKU84coj9wPn8Am6iQww==</latexit><latexit 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A3 = {3}<latexit sha1_base64="NdY91Z7nBx1SnRS8estTbjzY/GQ=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU1ItQ8eKxgv2AJJTNdtsu3eyG3Y1QQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZFyWcaeO6305pZXVtfaO8Wdna3tndq+4ftLVMFaEtIrlU3QhrypmgLcMMp91EURxHnHai8d3U7zxRpZkUj2aS0DDGQ8EGjGBjJT+77Z3nN0F2HuS9as2tuzOgZeIVpAYFmr3qV9CXJI2pMIRjrX3PTUyYYWUY4TSvBKmmCSZjPKS+pQLHVIfZ7OQcnViljwZS2RIGzdTfExmOtZ7Eke2MsRnpRW8q/uf5qRlchxkTSWqoIPNFg5QjI9H0f9RnihLDJ5Zgopi9FZERVpgYm1LFhuAtvrxM2md1z617Dxe1xmURRxmO4BhOwYMraMA9NKEFBCQ8wyu8OcZ5cd6dj3lrySlmDuEPnM8focqQxw==</latexit><latexit sha1_base64="NdY91Z7nBx1SnRS8estTbjzY/GQ=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU1ItQ8eKxgv2AJJTNdtsu3eyG3Y1QQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZFyWcaeO6305pZXVtfaO8Wdna3tndq+4ftLVMFaEtIrlU3QhrypmgLcMMp91EURxHnHai8d3U7zxRpZkUj2aS0DDGQ8EGjGBjJT+77Z3nN0F2HuS9as2tuzOgZeIVpAYFmr3qV9CXJI2pMIRjrX3PTUyYYWUY4TSvBKmmCSZjPKS+pQLHVIfZ7OQcnViljwZS2RIGzdTfExmOtZ7Eke2MsRnpRW8q/uf5qRlchxkTSWqoIPNFg5QjI9H0f9RnihLDJ5Zgopi9FZERVpgYm1LFhuAtvrxM2md1z617Dxe1xmURRxmO4BhOwYMraMA9NKEFBCQ8wyu8OcZ5cd6dj3lrySlmDuEPnM8focqQxw==</latexit><latexit sha1_base64="NdY91Z7nBx1SnRS8estTbjzY/GQ=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU1ItQ8eKxgv2AJJTNdtsu3eyG3Y1QQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZFyWcaeO6305pZXVtfaO8Wdna3tndq+4ftLVMFaEtIrlU3QhrypmgLcMMp91EURxHnHai8d3U7zxRpZkUj2aS0DDGQ8EGjGBjJT+77Z3nN0F2HuS9as2tuzOgZeIVpAYFmr3qV9CXJI2pMIRjrX3PTUyYYWUY4TSvBKmmCSZjPKS+pQLHVIfZ7OQcnViljwZS2RIGzdTfExmOtZ7Eke2MsRnpRW8q/uf5qRlchxkTSWqoIPNFg5QjI9H0f9RnihLDJ5Zgopi9FZERVpgYm1LFhuAtvrxM2md1z617Dxe1xmURRxmO4BhOwYMraMA9NKEFBCQ8wyu8OcZ5cd6dj3lrySlmDuEPnM8focqQxw==</latexit><latexit sha1_base64="NdY91Z7nBx1SnRS8estTbjzY/GQ=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU1ItQ8eKxgv2AJJTNdtsu3eyG3Y1QQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZFyWcaeO6305pZXVtfaO8Wdna3tndq+4ftLVMFaEtIrlU3QhrypmgLcMMp91EURxHnHai8d3U7zxRpZkUj2aS0DDGQ8EGjGBjJT+77Z3nN0F2HuS9as2tuzOgZeIVpAYFmr3qV9CXJI2pMIRjrX3PTUyYYWUY4TSvBKmmCSZjPKS+pQLHVIfZ7OQcnViljwZS2RIGzdTfExmOtZ7Eke2MsRnpRW8q/uf5qRlchxkTSWqoIPNFg5QjI9H0f9RnihLDJ5Zgopi9FZERVpgYm1LFhuAtvrxM2md1z617Dxe1xmURRxmO4BhOwYMraMA9NKEFBCQ8wyu8OcZ5cd6dj3lrySlmDuEPnM8focqQxw==</latexit>

A4 = {4}<latexit sha1_base64="XOSD6CexLkbM9tv+kGEbHWgum2M=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mkqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQu6p5b9x4ateZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHpNuQyQ==</latexit><latexit sha1_base64="XOSD6CexLkbM9tv+kGEbHWgum2M=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mkqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQu6p5b9x4ateZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHpNuQyQ==</latexit><latexit sha1_base64="XOSD6CexLkbM9tv+kGEbHWgum2M=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mkqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQu6p5b9x4ateZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHpNuQyQ==</latexit><latexit sha1_base64="XOSD6CexLkbM9tv+kGEbHWgum2M=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mkqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbfiO/CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQu6p5b9x4ateZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHpNuQyQ==</latexit>

d4 = 1<latexit sha1_base64="mXFggRNWuGQ0rQO0bQBdOWsLq1s=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1ItQ8OKxgmkLbSibzaZdutmE3YlQSn+DFw+KePUHefPfuG1z0NYHA4/3ZpiZF2ZSGHTdb6e0tr6xuVXeruzs7u0fVA+PWibNNeM+S2WqOyE1XArFfRQoeSfTnCah5O1wdDfz209cG5GqRxxnPEjoQIlYMIpW8qP+5a3Xr9bcujsHWSVeQWpQoNmvfvWilOUJV8gkNabruRkGE6pRMMmnlV5ueEbZiA5411JFE26CyfzYKTmzSkTiVNtSSObq74kJTYwZJ6HtTCgOzbI3E//zujnGN8FEqCxHrthiUZxLgimZfU4ioTlDObaEMi3srYQNqaYMbT4VG4K3/PIqaV3UPbfuPVzWGldFHGU4gVM4Bw+uoQH30AQfGAh4hld4c5Tz4rw7H4vWklPMHMMfOJ8/4N2OBQ==</latexit><latexit sha1_base64="mXFggRNWuGQ0rQO0bQBdOWsLq1s=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1ItQ8OKxgmkLbSibzaZdutmE3YlQSn+DFw+KePUHefPfuG1z0NYHA4/3ZpiZF2ZSGHTdb6e0tr6xuVXeruzs7u0fVA+PWibNNeM+S2WqOyE1XArFfRQoeSfTnCah5O1wdDfz209cG5GqRxxnPEjoQIlYMIpW8qP+5a3Xr9bcujsHWSVeQWpQoNmvfvWilOUJV8gkNabruRkGE6pRMMmnlV5ueEbZiA5411JFE26CyfzYKTmzSkTiVNtSSObq74kJTYwZJ6HtTCgOzbI3E//zujnGN8FEqCxHrthiUZxLgimZfU4ioTlDObaEMi3srYQNqaYMbT4VG4K3/PIqaV3UPbfuPVzWGldFHGU4gVM4Bw+uoQH30AQfGAh4hld4c5Tz4rw7H4vWklPMHMMfOJ8/4N2OBQ==</latexit><latexit sha1_base64="mXFggRNWuGQ0rQO0bQBdOWsLq1s=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1ItQ8OKxgmkLbSibzaZdutmE3YlQSn+DFw+KePUHefPfuG1z0NYHA4/3ZpiZF2ZSGHTdb6e0tr6xuVXeruzs7u0fVA+PWibNNeM+S2WqOyE1XArFfRQoeSfTnCah5O1wdDfz209cG5GqRxxnPEjoQIlYMIpW8qP+5a3Xr9bcujsHWSVeQWpQoNmvfvWilOUJV8gkNabruRkGE6pRMMmnlV5ueEbZiA5411JFE26CyfzYKTmzSkTiVNtSSObq74kJTYwZJ6HtTCgOzbI3E//zujnGN8FEqCxHrthiUZxLgimZfU4ioTlDObaEMi3srYQNqaYMbT4VG4K3/PIqaV3UPbfuPVzWGldFHGU4gVM4Bw+uoQH30AQfGAh4hld4c5Tz4rw7H4vWklPMHMMfOJ8/4N2OBQ==</latexit><latexit sha1_base64="mXFggRNWuGQ0rQO0bQBdOWsLq1s=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1ItQ8OKxgmkLbSibzaZdutmE3YlQSn+DFw+KePUHefPfuG1z0NYHA4/3ZpiZF2ZSGHTdb6e0tr6xuVXeruzs7u0fVA+PWibNNeM+S2WqOyE1XArFfRQoeSfTnCah5O1wdDfz209cG5GqRxxnPEjoQIlYMIpW8qP+5a3Xr9bcujsHWSVeQWpQoNmvfvWilOUJV8gkNabruRkGE6pRMMmnlV5ueEbZiA5411JFE26CyfzYKTmzSkTiVNtSSObq74kJTYwZJ6HtTCgOzbI3E//zujnGN8FEqCxHrthiUZxLgimZfU4ioTlDObaEMi3srYQNqaYMbT4VG4K3/PIqaV3UPbfuPVzWGldFHGU4gVM4Bw+uoQH30AQfGAh4hld4c5Tz4rw7H4vWklPMHMMfOJ8/4N2OBQ==</latexit>

d1 = 2<latexit sha1_base64="W6UvBpEI7GtHd5wGf5yXIk4+fio=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9SIUvHisYNpCG8pms2mXbjZhdyKU0t/gxYMiXv1B3vw3btsctPXBwOO9GWbmhZkUBl3321lb39jc2i7tlHf39g8OK0fHLZPmmnGfpTLVnZAaLoXiPgqUvJNpTpNQ8nY4upv57SeujUjVI44zHiR0oEQsGEUr+VHfu633K1W35s5BVolXkCoUaPYrX70oZXnCFTJJjel6bobBhGoUTPJpuZcbnlE2ogPetVTRhJtgMj92Ss6tEpE41bYUkrn6e2JCE2PGSWg7E4pDs+zNxP+8bo7xTTARKsuRK7ZYFOeSYEpmn5NIaM5Qji2hTAt7K2FDqilDm0/ZhuAtv7xKWvWa59a8h8tq46qIowSncAYX4ME1NOAemuADAwHP8ApvjnJenHfnY9G65hQzJ/AHzucP3c+OAw==</latexit><latexit sha1_base64="W6UvBpEI7GtHd5wGf5yXIk4+fio=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9SIUvHisYNpCG8pms2mXbjZhdyKU0t/gxYMiXv1B3vw3btsctPXBwOO9GWbmhZkUBl3321lb39jc2i7tlHf39g8OK0fHLZPmmnGfpTLVnZAaLoXiPgqUvJNpTpNQ8nY4upv57SeujUjVI44zHiR0oEQsGEUr+VHfu633K1W35s5BVolXkCoUaPYrX70oZXnCFTJJjel6bobBhGoUTPJpuZcbnlE2ogPetVTRhJtgMj92Ss6tEpE41bYUkrn6e2JCE2PGSWg7E4pDs+zNxP+8bo7xTTARKsuRK7ZYFOeSYEpmn5NIaM5Qji2hTAt7K2FDqilDm0/ZhuAtv7xKWvWa59a8h8tq46qIowSncAYX4ME1NOAemuADAwHP8ApvjnJenHfnY9G65hQzJ/AHzucP3c+OAw==</latexit><latexit sha1_base64="W6UvBpEI7GtHd5wGf5yXIk4+fio=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9SIUvHisYNpCG8pms2mXbjZhdyKU0t/gxYMiXv1B3vw3btsctPXBwOO9GWbmhZkUBl3321lb39jc2i7tlHf39g8OK0fHLZPmmnGfpTLVnZAaLoXiPgqUvJNpTpNQ8nY4upv57SeujUjVI44zHiR0oEQsGEUr+VHfu633K1W35s5BVolXkCoUaPYrX70oZXnCFTJJjel6bobBhGoUTPJpuZcbnlE2ogPetVTRhJtgMj92Ss6tEpE41bYUkrn6e2JCE2PGSWg7E4pDs+zNxP+8bo7xTTARKsuRK7ZYFOeSYEpmn5NIaM5Qji2hTAt7K2FDqilDm0/ZhuAtv7xKWvWa59a8h8tq46qIowSncAYX4ME1NOAemuADAwHP8ApvjnJenHfnY9G65hQzJ/AHzucP3c+OAw==</latexit><latexit sha1_base64="W6UvBpEI7GtHd5wGf5yXIk4+fio=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9SIUvHisYNpCG8pms2mXbjZhdyKU0t/gxYMiXv1B3vw3btsctPXBwOO9GWbmhZkUBl3321lb39jc2i7tlHf39g8OK0fHLZPmmnGfpTLVnZAaLoXiPgqUvJNpTpNQ8nY4upv57SeujUjVI44zHiR0oEQsGEUr+VHfu633K1W35s5BVolXkCoUaPYrX70oZXnCFTJJjel6bobBhGoUTPJpuZcbnlE2ogPetVTRhJtgMj92Ss6tEpE41bYUkrn6e2JCE2PGSWg7E4pDs+zNxP+8bo7xTTARKsuRK7ZYFOeSYEpmn5NIaM5Qji2hTAt7K2FDqilDm0/ZhuAtv7xKWvWa59a8h8tq46qIowSncAYX4ME1NOAemuADAwHP8ApvjnJenHfnY9G65hQzJ/AHzucP3c+OAw==</latexit>

d2 = 3<latexit sha1_base64="cWjunNJMMdmF+rQUPTu7EjV0a80=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqqBeh4MVjBdMW2lA2m027dLMbdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NYyU4T6RHKpuiHWlDNBfcMMp91UUZyEnHbC8d3M7zxRpZkUj2aS0iDBQ8FiRrCxkh8NGrcXg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY617npuaIMfKMMLptNLPNE0xGeMh7VkqcEJ1kM+PnaIzq0QolsqWMGiu/p7IcaL1JAltZ4LNSC97M/E/r5eZ+CbImUgzQwVZLIozjoxEs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQbdc+tew+XteZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzhvDjvzseiteQUM8fwB87nD+DZjgU=</latexit><latexit sha1_base64="cWjunNJMMdmF+rQUPTu7EjV0a80=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqqBeh4MVjBdMW2lA2m027dLMbdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NYyU4T6RHKpuiHWlDNBfcMMp91UUZyEnHbC8d3M7zxRpZkUj2aS0iDBQ8FiRrCxkh8NGrcXg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY617npuaIMfKMMLptNLPNE0xGeMh7VkqcEJ1kM+PnaIzq0QolsqWMGiu/p7IcaL1JAltZ4LNSC97M/E/r5eZ+CbImUgzQwVZLIozjoxEs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQbdc+tew+XteZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzhvDjvzseiteQUM8fwB87nD+DZjgU=</latexit><latexit sha1_base64="cWjunNJMMdmF+rQUPTu7EjV0a80=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqqBeh4MVjBdMW2lA2m027dLMbdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NYyU4T6RHKpuiHWlDNBfcMMp91UUZyEnHbC8d3M7zxRpZkUj2aS0iDBQ8FiRrCxkh8NGrcXg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY617npuaIMfKMMLptNLPNE0xGeMh7VkqcEJ1kM+PnaIzq0QolsqWMGiu/p7IcaL1JAltZ4LNSC97M/E/r5eZ+CbImUgzQwVZLIozjoxEs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQbdc+tew+XteZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzhvDjvzseiteQUM8fwB87nD+DZjgU=</latexit><latexit sha1_base64="cWjunNJMMdmF+rQUPTu7EjV0a80=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqqBeh4MVjBdMW2lA2m027dLMbdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NYyU4T6RHKpuiHWlDNBfcMMp91UUZyEnHbC8d3M7zxRpZkUj2aS0iDBQ8FiRrCxkh8NGrcXg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY617npuaIMfKMMLptNLPNE0xGeMh7VkqcEJ1kM+PnaIzq0QolsqWMGiu/p7IcaL1JAltZ4LNSC97M/E/r5eZ+CbImUgzQwVZLIozjoxEs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQbdc+tew+XteZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzhvDjvzseiteQUM8fwB87nD+DZjgU=</latexit>

d3 = 4<latexit sha1_base64="Dzvoa4vpObiWJLCDzizaOBHfcxw=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0qBeh4MVjBdMW2lA2m027dHcTdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NZJpgj1ScIT1Q2xppxJ6htmOO2mimIRctoJx3czv/NElWaJfDSTlAYCDyWLGcHGSn40uLxtDKo1t+7OgVaJV5AaFGgNql/9KCGZoNIQjrXueW5qghwrwwin00o/0zTFZIyHtGepxILqIJ8fO0VnVolQnChb0qC5+nsix0LriQhtp8BmpJe9mfif18tMfBPkTKaZoZIsFsUZRyZBs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQv6p5b9x4ateZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzpvDjvzseiteQUM8fwB87nD+Pjjgc=</latexit><latexit sha1_base64="Dzvoa4vpObiWJLCDzizaOBHfcxw=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0qBeh4MVjBdMW2lA2m027dHcTdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NZJpgj1ScIT1Q2xppxJ6htmOO2mimIRctoJx3czv/NElWaJfDSTlAYCDyWLGcHGSn40uLxtDKo1t+7OgVaJV5AaFGgNql/9KCGZoNIQjrXueW5qghwrwwin00o/0zTFZIyHtGepxILqIJ8fO0VnVolQnChb0qC5+nsix0LriQhtp8BmpJe9mfif18tMfBPkTKaZoZIsFsUZRyZBs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQv6p5b9x4ateZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzpvDjvzseiteQUM8fwB87nD+Pjjgc=</latexit><latexit sha1_base64="Dzvoa4vpObiWJLCDzizaOBHfcxw=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0qBeh4MVjBdMW2lA2m027dHcTdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NZJpgj1ScIT1Q2xppxJ6htmOO2mimIRctoJx3czv/NElWaJfDSTlAYCDyWLGcHGSn40uLxtDKo1t+7OgVaJV5AaFGgNql/9KCGZoNIQjrXueW5qghwrwwin00o/0zTFZIyHtGepxILqIJ8fO0VnVolQnChb0qC5+nsix0LriQhtp8BmpJe9mfif18tMfBPkTKaZoZIsFsUZRyZBs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQv6p5b9x4ateZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzpvDjvzseiteQUM8fwB87nD+Pjjgc=</latexit><latexit sha1_base64="Dzvoa4vpObiWJLCDzizaOBHfcxw=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0qBeh4MVjBdMW2lA2m027dHcTdjdCCf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXphypo3rfjultfWNza3ydmVnd2//oHp41NZJpgj1ScIT1Q2xppxJ6htmOO2mimIRctoJx3czv/NElWaJfDSTlAYCDyWLGcHGSn40uLxtDKo1t+7OgVaJV5AaFGgNql/9KCGZoNIQjrXueW5qghwrwwin00o/0zTFZIyHtGepxILqIJ8fO0VnVolQnChb0qC5+nsix0LriQhtp8BmpJe9mfif18tMfBPkTKaZoZIsFsUZRyZBs89RxBQlhk8swUQxeysiI6wwMTafig3BW355lbQv6p5b9x4ateZVEUcZTuAUzsGDa2jCPbTABwIMnuEV3hzpvDjvzseiteQUM8fwB87nD+Pjjgc=</latexit>

Figure: One optimal code is .{w1 + w2, w2 + w3, w3 + w4}

Index coding


Tx


A1 = {1}<latexit sha1_base64="RIuvxVxW/+gOOjb4zxD2Q/KzPeg=">AAAB8nicbVBNS8NAEN3Ur1q/qh69BIvgqWRF1ItQ8eKxgrWFJJTNdtMu3eyG3YlQQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZF6WCG/C8b6eysrq2vlHdrG1t7+zu1fcPHo3KNGUdqoTSvYgYJrhkHeAgWC/VjCSRYN1ofDv1u09MG67kA0xSFiZkKHnMKQEr+flN0cfXQY6Dol9veE1vBneZ4JI0UIl2v/4VDBTNEiaBCmKMj70Uwpxo4FSwohZkhqWEjsmQ+ZZKkjAT5rOTC/fEKgM3VtqWBHem/p7ISWLMJIlsZ0JgZBa9qfif52cQX4U5l2kGTNL5ojgTLih3+r874JpREBNLCNXc3urSEdGEgk2pZkPAiy8vk8ezJvaa+P680boo46iiI3SMThFGl6iF7lAbdRBFCj2jV/TmgPPivDsf89aKU84coj9wPn8AnBKQww==</latexit><latexit sha1_base64="RIuvxVxW/+gOOjb4zxD2Q/KzPeg=">AAAB8nicbVBNS8NAEN3Ur1q/qh69BIvgqWRF1ItQ8eKxgrWFJJTNdtMu3eyG3YlQQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZF6WCG/C8b6eysrq2vlHdrG1t7+zu1fcPHo3KNGUdqoTSvYgYJrhkHeAgWC/VjCSRYN1ofDv1u09MG67kA0xSFiZkKHnMKQEr+flN0cfXQY6Dol9veE1vBneZ4JI0UIl2v/4VDBTNEiaBCmKMj70Uwpxo4FSwohZkhqWEjsmQ+ZZKkjAT5rOTC/fEKgM3VtqWBHem/p7ISWLMJIlsZ0JgZBa9qfif52cQX4U5l2kGTNL5ojgTLih3+r874JpREBNLCNXc3urSEdGEgk2pZkPAiy8vk8ezJvaa+P680boo46iiI3SMThFGl6iF7lAbdRBFCj2jV/TmgPPivDsf89aKU84coj9wPn8AnBKQww==</latexit><latexit sha1_base64="RIuvxVxW/+gOOjb4zxD2Q/KzPeg=">AAAB8nicbVBNS8NAEN3Ur1q/qh69BIvgqWRF1ItQ8eKxgrWFJJTNdtMu3eyG3YlQQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZF6WCG/C8b6eysrq2vlHdrG1t7+zu1fcPHo3KNGUdqoTSvYgYJrhkHeAgWC/VjCSRYN1ofDv1u09MG67kA0xSFiZkKHnMKQEr+flN0cfXQY6Dol9veE1vBneZ4JI0UIl2v/4VDBTNEiaBCmKMj70Uwpxo4FSwohZkhqWEjsmQ+ZZKkjAT5rOTC/fEKgM3VtqWBHem/p7ISWLMJIlsZ0JgZBa9qfif52cQX4U5l2kGTNL5ojgTLih3+r874JpREBNLCNXc3urSEdGEgk2pZkPAiy8vk8ezJvaa+P680boo46iiI3SMThFGl6iF7lAbdRBFCj2jV/TmgPPivDsf89aKU84coj9wPn8AnBKQww==</latexit><latexit sha1_base64="RIuvxVxW/+gOOjb4zxD2Q/KzPeg=">AAAB8nicbVBNS8NAEN3Ur1q/qh69BIvgqWRF1ItQ8eKxgrWFJJTNdtMu3eyG3YlQQn6GFw+KePXXePPfuG1z0NYHA4/3ZpiZF6WCG/C8b6eysrq2vlHdrG1t7+zu1fcPHo3KNGUdqoTSvYgYJrhkHeAgWC/VjCSRYN1ofDv1u09MG67kA0xSFiZkKHnMKQEr+flN0cfXQY6Dol9veE1vBneZ4JI0UIl2v/4VDBTNEiaBCmKMj70Uwpxo4FSwohZkhqWEjsmQ+ZZKkjAT5rOTC/fEKgM3VtqWBHem/p7ISWLMJIlsZ0JgZBa9qfif52cQX4U5l2kGTNL5ojgTLih3+r874JpREBNLCNXc3urSEdGEgk2pZkPAiy8vk8ezJvaa+P680boo46iiI3SMThFGl6iF7lAbdRBFCj2jV/TmgPPivDsf89aKU84coj9wPn8AnBKQww==</latexit>

A2 = {2}<latexit sha1_base64="Cd8nxc7RsqYgpc/isAyfmaUkR6Q=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbvN+4CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQadc+tew8XteZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHnyKQxQ==</latexit><latexit sha1_base64="Cd8nxc7RsqYgpc/isAyfmaUkR6Q=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbvN+4CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQadc+tew8XteZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHnyKQxQ==</latexit><latexit sha1_base64="Cd8nxc7RsqYgpc/isAyfmaUkR6Q=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbvN+4CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQadc+tew8XteZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHnyKQxQ==</latexit><latexit sha1_base64="Cd8nxc7RsqYgpc/isAyfmaUkR6Q=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKqBeh4sVjBfsBSSib7bZdutkNuxOhhPwMLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oEN+C6305pbX1jc6u8XdnZ3ds/qB4edYxKNWVtqoTSvYgYJrhkbeAgWC/RjMSRYN1ocjfzu09MG67kI0wTFsZkJPmQUwJW8rPbvN+4CbJGkPerNbfuzoFXiVeQGirQ6le/goGiacwkUEGM8T03gTAjGjgVLK8EqWEJoRMyYr6lksTMhNn85ByfWWWAh0rbkoDn6u+JjMTGTOPIdsYExmbZm4n/eX4Kw+sw4zJJgUm6WDRMBQaFZ//jAdeMgphaQqjm9lZMx0QTCjalig3BW355lXQadc+tew8XteZlEUcZnaBTdI48dIWa6B61UBtRpNAzekVvDjgvzrvzsWgtOcXMMfoD5/MHnyKQxQ==</latexit>

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d2 2 {1, 3, 4}<latexit sha1_base64="V6oAwHhgptvgxUV93N7d7npFemU=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfSbLuVu7tQv6y135ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDDiR/Q==</latexit><latexit sha1_base64="V6oAwHhgptvgxUV93N7d7npFemU=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfSbLuVu7tQv6y135ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDDiR/Q==</latexit><latexit sha1_base64="V6oAwHhgptvgxUV93N7d7npFemU=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfSbLuVu7tQv6y135ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDDiR/Q==</latexit><latexit sha1_base64="V6oAwHhgptvgxUV93N7d7npFemU=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfSbLuVu7tQv6y135ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDDiR/Q==</latexit>

d3 2 {1, 2, 4}<latexit sha1_base64="btYAgnj9UVzRxxJje1Cn9GIPpR4=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfQvXcrd3Kk36y135ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDECR/Q==</latexit><latexit sha1_base64="btYAgnj9UVzRxxJje1Cn9GIPpR4=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfQvXcrd3Kk36y135ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDECR/Q==</latexit><latexit sha1_base64="btYAgnj9UVzRxxJje1Cn9GIPpR4=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfQvXcrd3Kk36y135ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDECR/Q==</latexit><latexit sha1_base64="btYAgnj9UVzRxxJje1Cn9GIPpR4=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfQvXcrd3Kk36y135ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDECR/Q==</latexit>

d4 2 {1, 2, 3}<latexit sha1_base64="mVmhnxcLNkcxkR6037hJ4YvAQcA=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfRbLuVu7tSb9Ut35ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDEqR/Q==</latexit><latexit sha1_base64="mVmhnxcLNkcxkR6037hJ4YvAQcA=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfRbLuVu7tSb9Ut35ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDEqR/Q==</latexit><latexit sha1_base64="mVmhnxcLNkcxkR6037hJ4YvAQcA=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfRbLuVu7tSb9Ut35ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDEqR/Q==</latexit><latexit sha1_base64="mVmhnxcLNkcxkR6037hJ4YvAQcA=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoJVgED6UktajHghePFewHNCFsNpt26WYTdjdCDf0lXjwo4tWf4s1/47bNQVsfDDzem2FmXpAyKpVtfxuljc2t7Z3ybmVv/+Cwah4d92SSCUy6OGGJGARIEkY56SqqGBmkgqA4YKQfTG7nfv+RCEkT/qCmKfFiNOI0ohgpLflmNfRbLuVu7tSb9Ut35ps1u2EvYK0TpyA1KNDxzS83THAWE64wQ1IOHTtVXo6EopiRWcXNJEkRnqARGWrKUUykly8On1nnWgmtKBG6uLIW6u+JHMVSTuNAd8ZIjeWqNxf/84aZim68nPI0U4Tj5aIoY5ZKrHkKVkgFwYpNNUFYUH2rhcdIIKx0VhUdgrP68jrpNRuO3XDuW7X2VRFHGU7hDC7AgWtowx10oAsYMniGV3gznowX4934WLaWjGLmBP7A+PwBDEqR/Q==</latexit>

Figure: and are optimal.{w1, w3} {w1 + w2, w3 + w4}

PICOD

15



Tx











PICOD

15



Tx











PICOD

15

Figure: is not a decentralized code. is decentralized code.

{w1 + w2, w3 + w4}{w1, w3}

Decentralized PICOD



Tx











PICOD

15

Figure: is not a decentralized code. is decentralized code.

{w1 + w2, w3 + w4}{w1, w3}

Decentralized PICOD


Tx










Secure PICOD

Figure: is secured code. is not secure code.

{w1 + w2, w3 + w4} {w1, w3}


Relevant past work

16


Index Coding• General index coding problem is open.• Index coding is equivalent to network coding[1].• There are index coding instances that require non-Shannon type of

inequalities[2] (non-negativity of conditional mutual information).• There are index coding instances where nonlinear schemes strictly outperforms

all linear schemes[3].

We do not solve PICOD by solving the corresponding family of IC.

[1] M. Effros, S. E. Rouayheb, M. Langberg, ”An Equivalence Between Network Coding and Index Coding,” IEEE Trans. Information Theory, May 2015. [2] H. Sun, S. A. Jafar, ”Index Coding Capacity: How Far Can One Go With Only Shannon Inequalities?” IEEE Trans. Information Theory, June 2015. [3] E. Lubetzky, U. Stav, ”Nonlinear Index Coding Outperforming the Linear Optimum,” IEEE Trans. Information Theory, Aug 2009.

17


PICOD(t) # of messages; maximum/minimum size of side information set; # of desired message per user.m = smax/smin = t =

• Past work • Achievability:

• Sufficiency of transmissions for PICOD(1) if each user has side information set of the same size [1].

• Sufficiency of transmissions for PICOD(t)[2].• Linear converse:

• Necessity of transmissions for linear scheme in PICOD(t) where transmitter only knows side information size range [1].

• Necessity of transmissions for linear scheme in a construction of PICOD(1)[2].

• Our contributions • For PICOD(1) with same size of side information at the users, one transmission satisfies at least

fraction of unsatisfied users in the system[3].

• Information theoretical optimality for some complete-S PICOD(t) and PICOD(1) with circular-arc side information[4,5,6].

[1] S. Brahma and C. Fragouli, ”Pliable index coding,” IEEE Trans. Information Theory, Nov 2015. [2] L. Song and C. Fragouli, ”A Polynomial-Time Algorithm for Pliable Index Coding,” IEEE Trans. Information Theory, Feb 2018. [3] T. Liu and D. Tuninetti, “Pliable Index Coding: Novel Lower Bound on the Fraction of Satisfied Clients with a Single Transmission and its Application,” ITW, 2016.[4] T. Liu and D. Tuninetti, “Information Theoretic Converse Proofs for Some PICOD Problems,” ITW, 2017. [5] T. Liu and D. Tuninetti, “An Information Theoretic Converse for the “Consecutive CompleteS” PICOD Problem,” ITW, 2018. [6] T. Liu and D. Tuninetti, “Tight Information Theoretic Converse Results for some Pliable Index Coding Problems,” IEEE Trans. on Information Theory, 2019.

O(log n)( |Ai | = s, ∀i)

O(t log(n) + log2(n))

min{smax + t, m − smin}[smin : smax]

log(n)

max{(1 −sm

) ms −1e(1 −

1s

)s−1,sm

} ≥ 1/e

18


Decentralized PICOD(t)• Past work (decentralized IC)

• IC with multiple transmitters/distributed IC[1]: capacity up to 3 messages.• Decentralized/embedded IC[2]: multiplicative gap 2 to the centralized version.• Data exchange problem[3]: users request all messages that are not in the side information.

• Our contributions[4,5] • Achievability scheme based on vector index coding and decentralized MDS code for

decentralized complete—S PICOD(t) and PICOD(1) with circular-arc side information.

[1] Y. Liu, P. Sadeghi, F. Arbabjolfaei, and Y.-H. Kim, “Capacity theorems for distributed index coding,” arXiv:1801.09063, 2018. [2] M. W. Alexandra Porter, “Embedded index coding,” Information Theory Workshop, 2019. arXiv:1904.02179. [3] N. Milosavljevic, S. Pawar, S. E. Rouayheb, M. Gastpar, and K. Ramchandran, ``Efficient algorithms for the data exchange problem,'' IEEE Trans. Information Theory, vol. 62, no. 4, pp. 1878 – 1896, April 2016. [4] T. Liu and D. Tuninetti, “Decentralized pliable index coding.” ISIT, 2019. [5] T. Liu and D. Tuninetti, “Decentralized Pliable Index Coding.” Submitted to IEEE Trans. on Information Theory.

19


Secure PICOD(1)• Past work[1]

• # of messages = # of users = ; side information size = s; side information shift = 1.

• Under linear encoding constraint• Unfeasible cases.• Optimality for .

• Our contributions[2,3] • Generalized model: # of messages = , side

information size = , side information shift = .• Information theoretical converse tight for some cases.• Universal achievable scheme that differ from

optimality by one transmission under the linear encoding constraint.

m

s > m/2

ms g

[1] Shanuja Sasi, B. Sundar Rajan, ``On Pliable Index Coding,'' ISIT 2019. [2] T. Liu and D. Tuninetti, “Private pliable index coding.” ITW, 2019. [3] T. Liu and D. Tuninetti, “Individually Secure Pliable Index Coding.” Under preparation.

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Figure: Circular shift side

20


Our contributions

21


Our contributions1. For PICOD(1) where all users have the same size of side information, there exists a scheme that

satisfies at least a constant fraction of unsatisfied users at each transmission [1]. Deterministic construction and better faction guarantee.

2. Information theoretical optimality for some complete-S PICOD(t) and PICOD(1) with circular-arc side information [2,3,4]. Novel converse bound rooted in combinatorics.

3. Information theoretical optimality for some decentralized complete-S PICOD(t) and PICOD(1) with circular-arc side information [5,6]. First to pose and study the problem.

4. Linear near optimality for individual message secure PICOD(1) with circular-arc side information [7,8]. Improved scheme and information theoretic optimality in some cases.

[1] T. Liu and D. Tuninetti, “Pliable Index Coding: Novel Lower Bound on the Fraction of Satisfied Clients with a Single Transmission and its Application,” ITW, 2016.[2] T. Liu and D. Tuninetti, “Information Theoretic Converse Proofs for Some PICOD Problems,” ITW, 2017.[3] T. Liu and D. Tuninetti, “An Information Theoretic Converse for the “Consecutive CompleteS” PICOD Problem,” ITW, 2018. [4] T. Liu and D. Tuninetti, “Tight Information Theoretic Converse Results for some Pliable Index Coding Problems,” IEEE Trans. on Information Theory, 2019. [5] T. Liu and D. Tuninetti, “Decentralized pliable index coding.” ISIT, 2019. [6] T. Liu and D. Tuninetti, “Decentralized Pliable Index Coding.” Submitted to IEEE Trans. on Information Theory. [7] T. Liu and D. Tuninetti, “Private pliable index coding.” ITW, 2019. [8] T. Liu and D. Tuninetti, “Individually Secure Pliable Index Coding.” Under preparation.

22


Proofs in a longer version on these slides. See also PICOD ieeexplore 8871209

d-PICOD arXiv 1904.05272s-PICOD arXiv 1904.04468


Future work• Combine our method with others (e.g., absent user[1], weighted alignment

chain[2]) for improved converse bounds.

• Understand fundamental differences between PICOD and decentralized or secure version.

• Secure vs Privacy[3] in PICOD.

[1] L. Ong, B. N. Vellambi, J. Kliewer, “Optimal-Rate Characterisation for Pliable Index Coding using Absent Receivers,” ISIT 2019. [2] Y. Liu and P. Sadeghi, “From Alignment to Acyclic Chains: Lexicographic Performance Bounds for Index Coding,” Allerton 2019. [3] M. Karmoose, L. Song, M. Cardone and C. Fragouli, Privacy in Index Coding: Improved Bounds and Coding Schemes, in: IEEE International Symposium on Information Theory (ISIT), 2018

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https://arxiv.org/search/cs?searchtype=author&query=Ong%2C+L

https://arxiv.org/search/cs?searchtype=author&query=Vellambi%2C+B+N

https://arxiv.org/search/cs?searchtype=author&query=Kliewer%2C+J


THANK YOU!

25

Documents

Pliable Index Coding - nicest.lab.uic.edu€¦ · Tang Liu, PhD thesis defense, Nov 19th 2019, UIC Broadcast channel with message side information • Broadcast nature of wireless