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Introduction to Network Coding Tuvi Etzion Lecture 11 Vector Linear Network Coding Vector Linear Network Coding Outline Fundamentals for vector network coding The combination network Vector network code vs. scalar network code 2 Multicast

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SLIDE 1

Tuvi Etzion

Vector Linear Network Coding Lecture 11

Introduction to Network Coding

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SLIDE 2

Vector Linear Network Coding

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Fundamentals for vector network coding The combination network

Outline

Vector network code vs. scalar network code

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Multicast Vector Network Coding

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The source has π’Š inputs (vectors of length 𝒖) and each one of the 𝑢 receivers demands all the inputs.

An edge 𝒇 will carry a linear combination

  • f the π’Š inputs. This combination is the

global network code. The information carried by 𝒇 is also a linear combination of the information on the in-coming edges of the parent vertex π’˜ of 𝒇. This linear combination is the local network code.

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SLIDE 4

Multicast Vector Network Coding

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Let π’—πŸ, π’—πŸ‘, … , π’—π’Š be the π’Š inputs of the source. To each edge 𝒇 there exist π’Š 𝒖 Γ— 𝒖 matrices π‘©πŸ(𝒇), π‘©πŸ‘(𝒇), … , π‘©π’Š(𝐟) over 𝔾𝒓. The edge 𝒇 will carry the linear combination π‘©πŸ 𝒇 π’—πŸ + π‘©πŸ‘ 𝒇 π’—πŸ‘ + β‹― + π‘©π’Š 𝒇 π’—π’Š A receiver 𝑺 has π’Š in-coming edges on each

  • ne there are π’Š 𝒖 Γ— 𝒖 matrices which are the

coefficients of the linear combinations.

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SLIDE 5

Multicast Vector Network Coding

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The edge 𝒇 will carry the linear combination π‘©πŸ 𝒇 π’—πŸ + π‘©πŸ‘ 𝒇 π’—πŸ‘ + β‹― + π‘©π’Š 𝒇 π’—π’Š A receiver 𝑺 has π’Š in-coming edges on each

  • ne there are π’Š 𝒖 Γ— 𝒖 matrices which are the

coefficients of the linear combinations. The receiver 𝑺 forms a π’Šπ’– Γ— π’Šπ’– matrix 𝑫 from which it finds the inputs. The matrix 𝑫 is the transfer matrix. To have a solution 𝑫 must be of full rank.

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SLIDE 6

Multicast Vector Network Coding

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The vertex π’˜ has β„“ in-coming edges and 𝒇 is an out-going edge of π’˜ with the following local linear combination in the scalar solution π’ƒπŸπœΉπŸ + π’ƒπŸ‘πœΉπŸ‘ + β‹― + π’ƒβ„“πœΉβ„“ where 𝒃𝒋 = πœ·π’Œπ’‹, 𝜷 primitive in 𝔾𝒓𝒖. Let π‘ͺ the companion matrix related to a primitive polynomial. Let 𝚫

𝐣 be the vector

representation of πœΉπ’‹. The edge 𝒇 will carry the following local linear combination in the vector solution π‘ͺπ’ŒπŸπš«πŸ + π‘ͺπ’ŒπŸ‘πš«

πŸ‘ + β‹― + π‘ͺπ’Œβ„“πš« β„“

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SLIDE 7

The Combination network

π‘Άπ’Š,𝒔,π’Š

A unique transmitter has π’Š messages and it has

  • ut-degree 𝒔, i.e., it is connected to 𝒔 vertices.

Each π’Š vertices of these 𝒔 vertices are connected to a receiver, i.e., a total of 𝒔

π’Š receivers.

π‘Άπ’Š,𝒔,π’Š is solvable if and only if there exists an 𝒔, π’“π’Š, 𝒔 βˆ’ π’Š + 𝟐 𝒓 code.

Theorem A network with three layers

Riis, Ahlswede 2006

7

The combination network

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The Combination network

A unique transmitter has πŸ‘ messages and it has

  • ut-degree 𝒔, i.e., it is connected to 𝒔 vertices.

Each πŸ‘ vertices of these 𝒔 vertices are connected to a receiver, i.e., a total of 𝒔

π’Š receivers.

π‘ΆπŸ‘,𝒔,πŸ‘ is solvable if and only if there exists an 𝒔, π’“πŸ‘, 𝒔 + 𝟐 𝒓 MDS code.

Theorem

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The combination network π‘ΆπŸ‘,𝒔,πŸ‘.

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The Combination network

The unique transmitter has πŸ‘ messages which are vectors π’š and 𝒛 of length 𝒖. There exists a code with 𝒔 βˆ’ πŸ‘ 𝒖 Γ— 𝒖 matrices π‘©πŸ, π‘©πŸ‘, … , π‘©π’”βˆ’πŸ‘ with rank 𝒖 and minimum rank distance 𝒖. A middle layer node receive a vector 𝑱 π‘©π’Œ β‹… π’š 𝒛 𝑼

  • r 𝑱 𝟏 β‹… π’š 𝒛 𝑼 or 𝟏 𝑱 β‹… π’š 𝒛 𝑼. A receiver gets

two such matrices with the related vector of length πŸ‘π’– and since any two such matrices form a πŸ‘π’– Γ— πŸ‘π’– matrix of full rank, it can obtains the two input vectors.

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The combination network π‘ΆπŸ‘,𝒔,πŸ‘.

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Rank-Metric Codes

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A 𝒍 Γ— 𝒏, 𝝕, 𝜺 𝒓 code satisfies 𝝕 ≀ 𝐧𝐣𝐨{𝒍 𝒏 βˆ’ 𝜺 + 𝟐 , 𝒏(𝒍 βˆ’ 𝜺 + 𝟐)}

Theorem

There exists a 𝒍 Γ— 𝒏, 𝝕, 𝜺 𝒓 code which satisfies 𝝕 = 𝐧𝐣𝐨{𝒍 𝒏 βˆ’ 𝜺 + 𝟐 , 𝒏(𝒍 βˆ’ 𝜺 + 𝟐)}

Theorem

If 𝑩 is an 𝒍 Γ— 𝒏 matrix then [ 𝑱 𝑩 ] is a generator matrix of a 𝒍-dimensional subspace of 𝔾𝒓

𝒏+𝒍.

Lemma

If 𝑫 is a 𝒍 Γ— 𝒏, 𝝕, 𝜺 𝒓 code then β„‚ = { ⟨ 𝑱 𝒀 ⟩: 𝒀 ∈ 𝑫 } is a code in 𝑯𝒓(𝒏 + 𝒍, 𝒍) with 𝒆𝑻 β„‚ = πŸ‘πœΊ.

Theorem

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SLIDE 11

Lifted Rank-Metric Codes

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If 𝑫 is a 𝒍 Γ— 𝒏, 𝝕, 𝜺 𝒓 code then β„‚ = { ⟨ 𝑱 𝒀 ⟩: 𝒀 ∈ 𝑫 } is a code in 𝑯𝒓(𝒏 + 𝒍, 𝒍) with 𝒆𝑻 β„‚ = πŸ‘πœΊ.

Theorem

The subspace ⟨ [ 𝑱 𝑩 ] ⟩ is called the lifting of 𝑩. The code β„‚ = { ⟨ 𝑱 𝒀 ⟩: 𝒀 ∈ 𝑫 } is the lifting of 𝑫.

𝒓𝒍(π’βˆ’π’) < 𝒐 𝒍 𝒓 < πŸ“ β‹… 𝒓𝒍(π’βˆ’π’) Theorem

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Network Coding and Related Combinatorial Structures

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A SHORT BREAK

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Combination Network + Links

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Combination network + links π‘Άπ’Š,𝒔,π’Š

βˆ—

The number of inputs is π’Š = πŸ‘β„“ = πŸ“ π’Š = πŸ“ The number of nodes in the middle layer is 𝒔. The number of receivers is 𝒔

πŸ‘ .

There are β„“ links from the source to each node in the middle layer. The is also a link from the source to each receiver. πŸ’ layers

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Combination Network + Links

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The number of inputs is π’Š = πŸ‘β„“ = πŸ“ The number of nodes in the middle layer is 𝒔. The number of receivers is 𝒔

πŸ‘ .

There are β„“ links from the source to each node in the middle layer. There is also a link from the source to each receiver. πŸ’ layers A receiver gets information from two middle layer nodes. There are β„“ links from a middle node to the receiver.

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Combination Network + Links

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Combination network + links π‘ΆπŸ“,𝒔,πŸ“

βˆ—

Each coding vector is of length πŸ“ and hence each receiver must obtain from the nodes

  • f middle layer a subspace whose dimension

is at least πŸ’. Thus, in an optimal solution each middle node should have a distinct πŸ‘-dimensional subspace. Scalar solution 𝔾𝒓𝒕

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Combination Network + Links

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Each coding vector is of length πŸ“ and hence each receiver must obtain from the nodes of middle layer a subspace whose dimension is at least πŸ’. Thus, in an

  • ptimal solution each middle node should

have a distinct πŸ‘-dimensional subspace. π‘ΆπŸ“,𝒔,πŸ“

βˆ—

scalar solution 𝔾𝒓𝒕 𝒔 ≀ πŸ“ πŸ‘ 𝒓𝒕 ≀ (𝒓𝒕

πŸ‘ + 𝒓𝒕 + 𝟐)(𝒓𝒕 πŸ‘ + 𝟐)

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Combination Network + Links

Let 𝑫 be a πŸ‘π’– Γ— πŸ‘π’–, 𝝕, 𝒖 𝒓 code. Each two links for the same node receive and send information of the form 𝑱 𝑩 β‹… π’š 𝒛 π’œ 𝒙 𝑼, where 𝑩 is a codeword

  • f 𝑫, 𝑱 is the πŸ‘π’– Γ— πŸ‘π’– identity matrix,

and (π’š 𝒛 π’œ 𝒙) is the four inputs vector. π‘ΆπŸ“,𝒔,πŸ“

βˆ—

vector solution 𝔾𝒓

𝝕 = πŸ‘π’–(𝒖 + 𝟐) 𝒔 ≀ π’“πŸ‘π’–(𝒖+𝟐)

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Combination Network + Links

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π‘ΆπŸ“,𝒔,πŸ“

βˆ—

scalar solution 𝔾𝒓𝒕 𝒔 ≀ πŸ“ πŸ‘ 𝒓𝒕 ≀ (𝒓𝒕

πŸ‘ + 𝒓𝒕 + 𝟐)(𝒓𝒕 πŸ‘ + 𝟐)

π‘ΆπŸ“,𝒔,πŸ“

βˆ—

vector solution 𝔾𝒓

𝒔 ≀ π’“πŸ‘π’–(𝒖+𝟐)

π’“πŸ‘π’–(𝒖+𝟐) ≀ (𝒓𝒕

πŸ‘ + 𝒓𝒕 + 𝟐)(𝒓𝒕 πŸ‘ + 𝟐)

𝒓𝒕~π’“π’–πŸ‘ πŸ‘

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SLIDE 19

Combination Network + Links

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Combination network + links π‘Άπ’Š,𝒔,π’Š

βˆ—

The number of inputs is π’Š = πŸ‘β„“ The number of nodes in the middle layer is 𝒔. The number of receivers is 𝒔

πŸ‘ .

There are β„“ links from the source to each node in the middle layer. There is also a link from the source to each receiver. πŸ’ layers

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Combination Network + Links

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The number of inputs is π’Š = πŸ‘β„“ The number of nodes in the middle layer is 𝒔. The number of receivers is 𝒔

πŸ‘ .

There are β„“ links from the source to each node in the middle layer. The is also a link from the source to each receiver. πŸ’ layers A receiver gets information from two middle layer nodes. There are β„“ links from a middle node to the receiver.

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SLIDE 21

Combination Network + Links

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Combination network + links π‘Άπ’Š,𝒔,π’Š

βˆ—

A coding vector is of length π’Š = πŸ‘β„“. Hence, each receiver must obtain from the middle layer nodes a subspace whose dimension is at least πŸ‘β„“ βˆ’ 𝟐. Thus, in an optimal solution two middle nodes should have two distinct β„“-dimensional subspaces whose intersection is at most an 𝟐-dimensional subspace. Scalar solution 𝔾𝒓𝒕

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Combination Network + Links

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A coding vector is of length π’Š = πŸ‘β„“. In an optimal solution two middle nodes should have two distinct β„“-dimensional subspaces whose intersection is at most an 𝟐-dimensional subspace. π‘Άπ’Š,𝒔,π’Š

βˆ—

scalar solution 𝔾𝒓𝒕

𝒔 ≲ 𝒓𝒕

πŸ‘β„“

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Combination Network + Links

Let 𝑫 be a ℓ𝒖 Γ— ℓ𝒖, 𝝕, β„“ βˆ’ 𝟐 𝒖 𝒓 code. Each β„“ links for the same node receive and send information of the form 𝑱 𝑩 β‹… π’šπŸ, … , π’šπŸ‘β„“ 𝑼, where 𝑩 is a codeword of 𝑫, 𝑱 is the ℓ𝒖 Γ— ℓ𝒖 identity matrix, and (π’šπŸ, … , π’šπŸ‘β„“) is the πŸ‘β„“ inputs vector. π‘Άπ’Š,𝒔,π’Š

βˆ—

vector solution 𝔾𝒓

𝝕 = ℓ𝒖(𝒖 + 𝟐) 𝒔 ≀ 𝒓ℓ𝒖(𝒖+𝟐)

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Combination Network + Links

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π‘Άπ’Š,𝒔,π’Š

βˆ—

scalar solution 𝔾𝒓𝒕

𝒔 ≲ 𝒓𝒕

πŸ‘β„“

π‘Άπ’Š,𝒔,π’Š

βˆ—

vector solution 𝔾𝒓

𝒔 ≀ 𝒓ℓ𝒖(𝒖+𝟐)

𝒓ℓ𝒖(𝒖+𝟐) ≀ 𝒓𝒕

πŸ‘β„“

𝒓𝒕~π’“π’–πŸ‘ πŸ‘

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SLIDE 25

Combination Network + Links

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Combination network + links π‘Άπ’Š,𝒔,π’Š

+

The number of inputs is π’Š = πŸ‘β„“ The number of nodes in the middle layer is 𝒔. The number of receivers is 𝒔

πŸ‘ .

There are β„“ links from the source to each node in the middle layer. There are β„“ βˆ’ 𝟐 links from the source to each receiver. πŸ’ layers

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Combination Network + Links

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The number of inputs is π’Š = πŸ‘β„“ The number of nodes in the middle layer is 𝒔. The number of receivers is 𝒔

πŸ‘ .

There are β„“ links from the source to each node in the middle layer. The are β„“ βˆ’ 𝟐 links from the source to each receiver. πŸ’ layers A receiver gets information from two middle layer nodes. There are β„“ links from a middle node to the receiver.

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SLIDE 27

Combination Network + Links

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Combination network + links π‘Άπ’Š,𝒔,π’Š

+

A coding vector is of length π’Š = πŸ‘β„“. Hence, each receiver must obtain from the middle layer nodes a subspace whose dimension is at least β„“ + 𝟐. Thus, in an optimal solution two middle nodes should have two distinct β„“-dimensional subspaces. Scalar solution 𝔾𝒓𝒕

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Combination Network + Links

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A coding vector is of length π’Š = πŸ‘β„“. In an optimal solution two middle nodes should have two distinct β„“-dimensional subspaces. π‘Άπ’Š,𝒔,π’Š

+

scalar solution 𝔾𝒓𝒕

𝒔 ≲ 𝒓𝒕

β„“πŸ‘

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Combination Network + Links

Let 𝑫 be a ℓ𝒖 Γ— ℓ𝒖, 𝝕, 𝒖 𝒓 code. Each β„“ links for the same node receive and send information of the form 𝑱 𝑩 β‹… π’šπŸ, … , π’šπŸ‘β„“ 𝑼, where 𝑩 is a codeword of 𝑫, 𝑱 is the ℓ𝒖 Γ— ℓ𝒖 identity matrix, and (π’šπŸ, … , π’šπŸ‘β„“) is the πŸ‘β„“ inputs vector. π‘Άπ’Š,𝒔,π’Š

+

vector solution 𝔾𝒓

𝝕 = ℓ𝒖((β„“ βˆ’ 𝟐)𝒖 + 𝟐) 𝒔 ≀ 𝒓ℓ𝒖((β„“βˆ’πŸ)𝒖+𝟐)

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SLIDE 30

Combination Network + Links

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π‘Άπ’Š,𝒔,π’Š

+

scalar solution 𝔾𝒓𝒕

𝒔 ≲ 𝒓𝒕

β„“πŸ‘

π‘Άπ’Š,𝒔,π’Š

+

vector solution 𝔾𝒓

𝒔 ≀ 𝒓ℓ𝒖((β„“βˆ’πŸ)𝒖+𝟐)

𝒓ℓ𝒖((β„“βˆ’πŸ)𝒖+𝟐) ≀ 𝒓𝒕

β„“πŸ‘

𝒓𝒕~𝒓(β„“βˆ’πŸ)π’–πŸ‘ β„“

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SLIDE 31

Vector Linear Network Coding

31

And how can we have even more advantage in the alphabet with vector network coding for the same networks by using subspace codes?

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SLIDE 32

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END OF LECTURE 11

Introduction to Network Coding