Lecture 11 Vector Linear Network Coding Vector Linear Network - PowerPoint PPT Presentation

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 Vector Network Coding The source has 𝒊 inputs (vectors of length 𝒖 ) and each one of the 𝑶 receivers demands all the inputs. An edge 𝒇 will carry a linear combination of the 𝒊 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 𝒘 of 𝒇 . This linear combination is the local network code. 3

Multicast Vector Network Coding Let 𝒗 𝟐 , 𝒗 𝟑 , … , 𝒗 𝒊 be the 𝒊 inputs of the source. To each edge 𝒇 there exist 𝒊 𝒖 × 𝒖 matrices 𝑩 𝟐 (𝒇), 𝑩 𝟑 (𝒇), … , 𝑩 𝒊 (𝐟) over 𝔾 𝒓 . The edge 𝒇 will carry the linear combination 𝑩 𝟐 𝒇 𝒗 𝟐 + 𝑩 𝟑 𝒇 𝒗 𝟑 + ⋯ + 𝑩 𝒊 𝒇 𝒗 𝒊 A receiver 𝑺 has 𝒊 in-coming edges on each one there are 𝒊 𝒖 × 𝒖 matrices which are the coefficients of the linear combinations. 4

Multicast Vector Network Coding The edge 𝒇 will carry the linear combination 𝑩 𝟐 𝒇 𝒗 𝟐 + 𝑩 𝟑 𝒇 𝒗 𝟑 + ⋯ + 𝑩 𝒊 𝒇 𝒗 𝒊 A receiver 𝑺 has 𝒊 in-coming edges on each one there are 𝒊 𝒖 × 𝒖 matrices which are the coefficients of the linear combinations. The receiver 𝑺 forms a 𝒊𝒖 × 𝒊𝒖 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. 5

Multicast Vector Network Coding The vertex 𝒘 has ℓ in-coming edges and 𝒇 is an out-going edge of 𝒘 with the following local linear combination in the scalar solution 𝒃 𝟐 𝜹 𝟐 + 𝒃 𝟑 𝜹 𝟑 + ⋯ + 𝒃 ℓ 𝜹 ℓ where 𝒃 𝒋 = 𝜷 𝒌 𝒋 , 𝜷 primitive in 𝔾 𝒓 𝒖 . Let 𝑪 the companion matrix related to a primitive polynomial. Let 𝚫 𝐣 be the vector representation of 𝜹 𝒋 . The edge 𝒇 will carry the following local linear combination in the vector solution 𝑪 𝒌 𝟐 𝚫 𝟐 + 𝑪 𝒌 𝟑 𝚫 ℓ 𝟑 + ⋯ + 𝑪 𝒌 ℓ 𝚫 6

The Combination network A network with three layers 𝑶 𝒊,𝒔,𝒊 A unique transmitter has 𝒊 messages and it has out-degree 𝒔 , i.e., it is connected to 𝒔 vertices. Each 𝒊 vertices of these 𝒔 vertices are connected to a receiver, i.e., a total of 𝒔 𝒊 receivers. The combination network Riis, Ahlswede 2006 Theorem 𝑶 𝒊,𝒔,𝒊 is solvable if and only if there exists an 𝒔, 𝒓 𝒊 , 𝒔 − 𝒊 + 𝟐 𝒓 code. 7

The Combination network The combination network 𝑶 𝟑,𝒔,𝟑 . A unique transmitter has 𝟑 messages and it has out-degree 𝒔 , i.e., it is connected to 𝒔 vertices. Each 𝟑 vertices of these 𝒔 vertices are connected to a receiver, i.e., a total of 𝒔 𝒊 receivers. Theorem 𝑶 𝟑,𝒔,𝟑 is solvable if and only if there exists an 𝒔, 𝒓 𝟑 , 𝒔 + 𝟐 𝒓 MDS code. 8

The Combination network The combination network 𝑶 𝟑,𝒔,𝟑 . The unique transmitter has 𝟑 messages which are vectors 𝒚 and 𝒛 of length 𝒖 . There exists a code with 𝒔 − 𝟑 𝒖 × 𝒖 matrices 𝑩 𝟐 , 𝑩 𝟑 , … , 𝑩 𝒔−𝟑 with rank 𝒖 and minimum rank distance 𝒖 . A middle layer node receive a vector 𝑱 𝑩 𝒌 ⋅ 𝒚 𝒛 𝑼 or 𝑱 𝟏 ⋅ 𝒚 𝒛 𝑼 or 𝟏 𝑱 ⋅ 𝒚 𝒛 𝑼 . A receiver gets two such matrices with the related vector of length 𝟑𝒖 and since any two such matrices form a 𝟑𝒖 × 𝟑𝒖 matrix of full rank, it can obtains the two input vectors. 9

Rank-Metric Codes A 𝒍 × 𝒏, 𝝕, 𝜺 𝒓 code satisfies Theorem 𝝕 ≤ 𝐧𝐣𝐨{𝒍 𝒏 − 𝜺 + 𝟐 , 𝒏(𝒍 − 𝜺 + 𝟐)} Theorem There exists a 𝒍 × 𝒏, 𝝕, 𝜺 𝒓 code which satisfies 𝝕 = 𝐧𝐣𝐨{𝒍 𝒏 − 𝜺 + 𝟐 , 𝒏(𝒍 − 𝜺 + 𝟐)} Lemma If 𝑩 is an 𝒍 × 𝒏 matrix then [ 𝑱 𝑩 ] is a generator matrix of a 𝒍 -dimensional subspace of 𝔾 𝒓 𝒏+𝒍 . Theorem If 𝑫 is a 𝒍 × 𝒏, 𝝕, 𝜺 𝒓 code then ℂ = { ⟨ 𝑱 𝒀 ⟩: 𝒀 ∈ 𝑫 } is a code in 𝑯 𝒓 (𝒏 + 𝒍, 𝒍) with 𝒆 𝑻 ℂ = 𝟑𝜺 . 10

Lifted Rank-Metric Codes Theorem If 𝑫 is a 𝒍 × 𝒏, 𝝕, 𝜺 𝒓 code then ℂ = { ⟨ 𝑱 𝒀 ⟩: 𝒀 ∈ 𝑫 } is a code in 𝑯 𝒓 (𝒏 + 𝒍, 𝒍) with 𝒆 𝑻 ℂ = 𝟑𝜺 . The subspace ⟨ [ 𝑱 𝑩 ] ⟩ is called the lifting of 𝑩 . The code ℂ = { ⟨ 𝑱 𝒀 ⟩: 𝒀 ∈ 𝑫 } is the lifting of 𝑫 . 𝒓 𝒍(𝒐−𝒍) < 𝒐 < 𝟓 ⋅ 𝒓 𝒍(𝒐−𝒍) Theorem 𝒍 𝒓 11

Network Coding and Related Combinatorial Structures A SHORT BREAK 12

Combination Network + Links Combination network + links 𝑶 𝒊,𝒔,𝒊 ∗ 𝒊 = 𝟓 The number of inputs is 𝒊 = 𝟑ℓ = 𝟓 𝟒 layers The number of nodes in the middle layer is 𝒔 . The number of receivers is 𝒔 𝟑 . There are ℓ links from the source to each node in the middle layer. The is also a link from the source to each receiver. 13

Combination Network + Links The number of inputs is 𝒊 = 𝟑ℓ = 𝟓 𝟒 layers The number of nodes in the middle layer is 𝒔 . The number of receivers is 𝒔 𝟑 . There are ℓ links from the source to each node in the middle layer. There is also a link from the source to each receiver. A receiver gets information from two middle layer nodes. There are ℓ links from a middle node to the receiver. 14

Combination Network + Links Combination network + links 𝑶 𝟓,𝒔,𝟓 ∗ Scalar solution 𝔾 𝒓 𝒕 Each coding vector is of length 𝟓 and hence each receiver must obtain from the nodes of middle layer a subspace whose dimension is at least 𝟒 . Thus, in an optimal solution each middle node should have a distinct 𝟑 -dimensional subspace. 15

Combination Network + Links scalar solution 𝔾 𝒓 𝒕 ∗ 𝑶 𝟓,𝒔,𝟓 Each coding vector is of length 𝟓 and hence each receiver must obtain from the nodes of middle layer a subspace whose dimension is at least 𝟒 . Thus, in an optimal solution each middle node should have a distinct 𝟑 -dimensional subspace. 𝒔 ≤ 𝟓 𝟑 + 𝒓 𝒕 + 𝟐)(𝒓 𝒕 𝟑 + 𝟐) ≤ (𝒓 𝒕 𝟑 𝒓 𝒕 16

Combination Network + Links vector solution 𝔾 𝒓 ∗ 𝑶 𝟓,𝒔,𝟓 Let 𝑫 be a 𝟑𝒖 × 𝟑𝒖, 𝝕, 𝒖 𝒓 code. Each two links for the same node receive and send information of the form 𝑱 𝑩 ⋅ 𝒚 𝒛 𝒜 𝒙 𝑼 , where 𝑩 is a codeword of 𝑫 , 𝑱 is the 𝟑𝒖 × 𝟑𝒖 identity matrix, and (𝒚 𝒛 𝒜 𝒙) is the four inputs vector. 𝒔 ≤ 𝒓 𝟑𝒖(𝒖+𝟐) 𝝕 = 𝟑𝒖(𝒖 + 𝟐)

Combination Network + Links scalar solution 𝔾 𝒓 𝒕 ∗ 𝑶 𝟓,𝒔,𝟓 𝒔 ≤ 𝟓 𝟑 + 𝒓 𝒕 + 𝟐)(𝒓 𝒕 𝟑 + 𝟐) ≤ (𝒓 𝒕 𝟑 𝒓 𝒕 ∗ vector solution 𝔾 𝒓 𝑶 𝟓,𝒔,𝟓 𝒔 ≤ 𝒓 𝟑𝒖(𝒖+𝟐) 𝒓 𝟑𝒖(𝒖+𝟐) ≤ (𝒓 𝒕 𝟑 + 𝒓 𝒕 + 𝟐)(𝒓 𝒕 𝟑 + 𝟐) 𝒓 𝒕 ~𝒓 𝒖 𝟑 𝟑 18

Combination Network + Links Combination network + links 𝑶 𝒊,𝒔,𝒊 ∗ The number of inputs is 𝒊 = 𝟑ℓ 𝟒 layers The number of nodes in the middle layer is 𝒔 . The number of receivers is 𝒔 𝟑 . There are ℓ links from the source to each node in the middle layer. There is also a link from the source to each receiver. 19

Combination Network + Links The number of inputs is 𝒊 = 𝟑ℓ 𝟒 layers The number of nodes in the middle layer is 𝒔 . The number of receivers is 𝒔 𝟑 . There are ℓ links from the source to each node in the middle layer. The is also a link from the source to each receiver. A receiver gets information from two middle layer nodes. There are ℓ links from a middle node to the receiver. 20

Combination Network + Links Combination network + links 𝑶 𝒊,𝒔,𝒊 ∗ Scalar solution 𝔾 𝒓 𝒕 A coding vector is of length 𝒊 = 𝟑ℓ . Hence, each receiver must obtain from the middle layer nodes a subspace whose dimension is at least 𝟑ℓ − 𝟐 . Thus, in an optimal solution two middle nodes should have two distinct ℓ -dimensional subspaces whose intersection is at most an 𝟐 -dimensional subspace. 21

Combination Network + Links ∗ scalar solution 𝔾 𝒓 𝒕 𝑶 𝒊,𝒔,𝒊 A coding vector is of length 𝒊 = 𝟑ℓ . In an optimal solution two middle nodes should have two distinct ℓ -dimensional subspaces whose intersection is at most an 𝟐 -dimensional subspace. 𝟑ℓ 𝒔 ≲ 𝒓 𝒕 22

Combination Network + Links ∗ vector solution 𝔾 𝒓 𝑶 𝒊,𝒔,𝒊 Let 𝑫 be a ℓ𝒖 × ℓ𝒖, 𝝕, ℓ − 𝟐 𝒖 𝒓 code. Each ℓ links for the same node receive and send information of the form 𝑱 𝑩 ⋅ 𝒚 𝟐 , … , 𝒚 𝟑ℓ 𝑼 , where 𝑩 is a codeword of 𝑫 , 𝑱 is the ℓ𝒖 × ℓ𝒖 identity matrix, and (𝒚 𝟐 , … , 𝒚 𝟑ℓ ) is the 𝟑ℓ inputs vector. 𝝕 = ℓ𝒖(𝒖 + 𝟐) 𝒔 ≤ 𝒓 ℓ𝒖(𝒖+𝟐)