Network Coding: Basic Theory

1
Network Coding Basic Theory

• Raymond W. Yeung
• The Chinese University of Hong Kong

2
Springer, 2008 Preprint available
at http//iest2.ie.cuhk.edu.hk/whyeung/book2/
3
Outline

• Introduction
• Linear Network Coding Acyclic Networks
• Linear Network Codes
• Existence and Construction
• Generic Network Codes
• Random Network Coding
• Linear Network Coding Cyclic Networks
• Convolutional Network Codes
• Decoding of Convolutional Network Codes

4
Introduction
5
Multicast vs Broadcast

• Broadcast
• One-to-many communication without specific
  receiver addresses
• Multicast
• One-to-many communication with specific receiver
  addresses

6
What is Network Coding?

• The traditional approach (store-and-forward) to
  multicasting is to find an multicast tree in the
  network.
• The nodes acting as switches that route and/or
  replicate information.
• Network coding is a theory which changes the way
  we think of transmitting information in a network.

7
What is Network Coding?

• In the paradigm of network coding, nodes can
  encode information received from the input links
  before it is transmitted on the output links.
• Switching is a special case of network coding.

8
Butterfly Network I
9
Advantage of Network Coding

• 9 instead of 10 bits need to be transmitted in
  order to multicast 2 bits.
• If only 1 bit can be transmitted from node 3 to
  node 4, only 1.5 bits on the average can be
  multicast.

10
An Network Coding Example with Two Sources
11
Butterfly Network II
12
Wireless and Satellite Communications

• A node is a broadcast node by the nature of
  wireless/satellite communication.
• At a node
• all the output channels have the same capacity
• the same symbol is sent on each of the output
  channels.

13
A Graph-Theoretic Representation
14
Butterfly Network II Revisited
t2
t1
15
Wireless/Satellite Application
50 saving for downlink bandwidth!
16
Two Themes of Network Coding

• When there is 1 source to be multicast in a
  network, store-and-forward may fail to optimize
  bandwidth.
• When there are 2 or more independent sources to
  be transmitted in a network (even for unicast),
  store-and-forward may fail to optimize bandwidth.
• In short, Information is NOT a commodity!

17
Why Surprising?

• Classical information theory (C.E. Shannon, 1948)
  tells us that for point-to-point communication,
  all information can ultimately be represented as
  independent (i.i.d.) bits through data
  compression.
• Independent bits cannot be further compressed
  commodity.

18
Why Surprising?

• The folklore was that the commodity view for
  information continues to apply in networks.
• For a long time, processing of information at the
  intermediate nodes in a network for the purpose
  of transmission has not been considered.

19
Applications of Network Coding

• Computer Networks
• Wireless/Satellite Communications
• Distributed information storage/dissemination
  (P2P)
• Network Error Correction
• Secure Network Transmission
• Mathematics Matroid, Polynomial Equations, Ring
• Quantum Information
• Biology

20
Single-Source Linear Network Coding

• Acyclic Networks

21
Model of a Point-to-Point Network

• A network is represented by a graph G (V,E)
  with node set V and edge (channel) set E.
• A symbol from an alphabet F can be transmitted on
  each channel.
• There can be multiple edges between a pair of
  nodes.

22
Single-Source Network Coding

• The source node s generates an information vector
  
• x (x1 x2 x?) ? F?.
• What is the condition for a node t to be able to
  receive the information vector x?

23
Max-Flow Min-Cut Theorem

• Consider a graph G (V,E).
• There is a source node s and a sink node t in set
  V.
• A cut in the graph G is a partition of the node
  set V into two subsets V1 and V2 such that s is
  in V1 and t is in V2.
• The capacity of a cut is the total number of
  edges across the cut from V1 to V2.

24
Max-Flow Min-Cut Theorem

• The max-flow from s to t, denoted by maxflow(t),
  is the maximum rate at which a fluid (e.g.,
  water) can flow from s to t provided that the
  rate of flow in each edge does not exceed 1 and
  the flow conserves at each node.
• By the Max-Flow Min-Cut Theorem, maxflow(t) is
  equal to the minimum value of the capacity of a
  cut in G.

25
The Max-Flow Bound

• What is the condition for a node t to be able to
  receive the information vector
• x (x1 x2 x?) ?
• Obviously, if maxflow(T) lt ?, then t cannot
  possibly receive x.

26
The Basic Results

• If network coding is allowed, a node t can
  receive the information vector x iff
• maxflow(t) k
• i.e., the max-flow bound can be achieved
  simultaneously by all such nodes t. (Ahlswede et
  al. 00)
• Moreover, this can be achieved by linear network
  coding for a sufficiently large base field. (Li,
  Y and Cai, Koetter and Medard, 03).

27
Linear Network Codes
28
Acyclic Networks

• The following type of networks is considered
• Cycle-free.
• There is only one information source in the whole
  network, generated at the node s.
• Transmission is assumed to be instantaneous.
• The max-flow bound can be achieved by linear
  network coding to different extents.

29
Upstream-to-Downstream Order

• The nodes of an acyclic graph can be ordered in
  an upstream-to-downstream order (ancesterol
  order).
• When the nodes encode according to such an order,
  whenever a node encodes, all the information
  needed would have already been received on the
  input channel of that node.

30
Example
s
An upsteam-to-downstream order S, 2, 1, 3, 4,
t2, t1
1
2
3
4
t2
t1
31
Notation

• Let F be a finite field.
• The information source generates a vector
• x (x1 x2 x?) in F?.
• The value of the max-flow from s to a node t is
  denoted by maxflow(t).
• In(t) denotes the set of edges incoming at t.
• Out(t) denotes the set of edges outgoing from t.
• By convention, we install ? imaginary channels
  ending at the source node s.

32
Local Description

• A pair (d,e) of channels is called an adjacent
  pair if there exists a node t with d ? In(t) and
  e ? Out(t).
• Definition 19.6. An ?-dimensional linear network
  code on an acyclic network consists of a scalar
  kd,e, called the local encoding kernel, for every
  adjacent pair of channels (d,e).
• The local encoding kernel at the node t refers to
  the In(t)?Out(t) matrix Kt
  kd,ed?In(t),e?Out(t).

33
Local Description (cont.)

• Let ye be the symbol sent on the channel e.
• The local input-output relation at a node t is
  given by
• ye ?d?In(t) kd,e yd
• for all e ? Out(t), i.e., an output symbol from
  t is a linear combination of the input symbols at
  t.

34
Linear Mapping

• Let f F? (row vector) ? F. If f is linear, then
  there exists a column vector a ? F? such that
• f(x) xa
• for all row vectors x ? F?.

35
Global Encoding Kernel

• Given the local description, the code maps the
  information vector x to each symbol ye sent on
  the channel e.
• Obviously, such a mapping is linear.
• Therefore, ye xfe for some column vector fe ?
  F?.
• The vector fe, called the global encoding kernel
  for the channel e, is like a transfer function.
• Analogous to a column of a generator matrix of a
  block code.

36
Global Description

• Definition 19.7.
• An ?-dimensional linear network code consists of
• a scalar kd,e for every adjacent pair of channels
  (d,e)
• an ?-dimensional column vector fe for every
  channel e
• such that
• fe ?d?In(t) kd,e fd for e ? Out(t).
• The vectors fe corresponding to the ? imaginary
  channels e ? In(s) form the standard basis of the
  vector space F?.

37

• The global description of a linear network code
  incorporates the local description.
• The relation fe ?d?In(t) kd,e fd for e ? Out(t)
  
• says that the global encoding kernel of an
  output channel at t is a linear combination of
  the global encoding kernels of the input channels
  at t.
• From this, we have
• ye xfe x(?d?In(t) kd,e fd)
• ?d?In(t) kd,e (xfd)
• ?d?In(t) kd,e yd
• as in the local description.

38
Example
39
This shows the same network with the local
encoding kernels being indeterminates.
40
Desirable Properties of a Linear Network Code

• Definition 19.12. An ?-dimensional linear network
  code qualifies as a
• Linear multicast
• dim(Vt) ? for every non-source node t with
  maxflow(t) ? ?.
• Linear broadcast
• dim(Vt) min?, maxflow(t) for every
  non-source node t.
• Linear dispersion
• dim(VT) min?, maxflow(T) for every
  collection T of non-source nodes.
• Notation Vt ?fe e ? In(t)? VT ??t?T Vt?

41
Linear Multicast

• For every node t, if maxflow(t) ?, then t
  receives the whole vector space so that the
  information vector x can be recovered.
• If maxflow(t) lt ?, possibly no useful information
  can be recovered.
• A linear network code may not be a linear
  multicast.

42
Linear Broadcast

• For every node t, if maxflow(t) ?, then t
  receives the whole vector space so that the
  information vector x can be recovered.
• If maxflow(t) lt ?, then t receives maxflow(t)
  dimensions, but which dimensions to be received
  is not guaranteed.
• A linear broadcast is stronger than a linear
  multicast.

43
Application of Linear Broadcast

• A natural extension of linear multicast
• Can be useful for multi-rate network coding (Fong
  and Y 06)

44
Linear Dispersion

• For every collection of nodes T, if maxflow(T)
  ?, then T receives the whole vector space so that
  the information vector x can be recovered.
• If maxflow(T) lt ?, then T receives maxflow(t)
  dimensions.
• A linear dispersion is stronger than a linear
  broadcast.

45
Application of Linear Dispersion

• A collection of nodes, each of which has maxflow
  lt ?, can be pooled together to receive the whole
  information vector x.
• Facilitates network expansion.

46

• Not multicast
• Multicast but not broadcast
• Broadcast but not dispersion
• dispersion

47
Example

• The linear network code in the next slide is a
  linear dispersion regardless of the choice of the
  field F.
• Hence, it is also a linear broadcast and a linear
  multicast.

48
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49
Example

• The more general linear network code in the next
  slide is a linear multicast when
• ftw and fuw are linearly independent.
• fty and fxy are linearly independent.
• fuz and fwz are linearly independent.
• Equivalently, the criterion says that s, t, u, y,
  z, nr ? pq, and npsw nrux ? pnsw ? pquw are all
  nonzero.
• The network code in the previous slide is the
  special case with n r s t u v w x
  y z 1 and p q 0.

50
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51
Implementation of Linear Network Codes

• The global encoding kernels must be known at a
  sink node, which can be delivered on the fly (Ho
  et al. 2003)
• Consider using the same linear network code n gt ?
  times.
• Send an identity matrix through during the
  first ? time slots.
• Very similar to fountain code.
• If n gtgt ?, the overhead is small.

52
Implementation of Linear Network Codes
Notation F? fee??
53
Existence and Construction
54
The Effect of Field Size

• To construct a linear network code with desirable
  properties, the field F often needs to be
  sufficient large.
• When the field size is small, certain polynomial
  equations are unavoidable. E.g., zp - z 0 for
  all z ? GF(p).
• The next slide shows a network on which there
  exists a ternary linear multicast but not a
  binary linear multicast.
• This is similar to the construction of a
  Vandermonde matrix.

55
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56
Lemma 19.17

• Let g(z1, z2, , zn) be a nonzero polynomial
  with coefficients in a field F. If F is greater
  than the degree of g in every zj, then there
  exist a1, a2, , an ? F such that
• g(a1, a2, , an) ? 0.
• Proof By induction on n and invoke the fact that
  an nth degree polynomial cannot have more than n
  roots in any field (unique factorization).

57
Existence of Linear Multicast

• Theorem 19.20 (Koetter-Medard 03)
• There exists a linear multicast on an acyclic
  network for sufficiently large base field F.

58
Example

• Refer to the next slide.
• Lw, Ly, and Lz are the matrices of the global
  encoding kernels received at nodes w, y, and z,
  resp.
• Clearly, det(LW), det(LY), det(LZ) are not the
  zero polynomial, or equivalently,
• P(n,p,q,) det(LW)det(LY)det(LZ)
• is not the zero polynomial.
• When F is sufficiently large, by Lemma 19.17, it
  is possible to assign the global encoding kernels
  so that P(n,p,q,) is not evaluated to 0.

59
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60
Proof of Theorem 19.20

• Let the local encoding kernels be indeterminates
  and from them determine the global encoding
  kernels accordingly.
• For any node t such that maxflow(t) ?, there
  exist ? edge-disjoint paths connecting the ?
  imaginary channels in In(s) to t.
• Let Lt be the matrix of the global encoding
  kernels of the channels of these paths ending at
  t.
• Claim det(Lt) is not the zero polynomial.

61

• Set the local encoding kernels to emulate the
  routing of ? symbols from s to t.
• With these settings of the local encoding
  kernels, Lt is evaluated to the ??? identity
  matrix and det(Lt) is evaluated to 1.
• Therefore, det(Lt) cannot be the zero polynomial.
• Consider ?Tmaxflow(t)? det(Lt), which is not
  the zero polynomial.
• For a sufficient large base field F, it is
  possible to set the local encoding kernels so
  that this polynomial is not evaluated to 0.
• The feasible local encoding kernels can in
  principle be found by exhaustive search over the
  base field F, but not in polynomial time.

62
Random Network Coding(Ho et al.)

• If F is very large, by choosing the local
  encoding kernels randomly from F, most likely the
  linear network code is a linear multicast and can
  be decoded.
• Code construction can be done independent of the
  network topology.

63
Random Network Coding

• Advantage
• Useful when the network is unknown, e.g., mobile
  ad hoc network (MANET).
• Disadvantage
• Need to use a large base field.
• Need to use the code repeatedly.

64
Jaggi-Sanders Algorithm

• Deterministic construction of linear multicast.
• The field size needs to be at least ?, the number
  of sink nodes.
• The complexity is proportional to E?.
• It is clear that if F is very large, a linear
  multicast can be constructed with very high
  probability by randomly picking the global
  encoding kernels.

65

• Two edge-disjoint paths to Y
• PY,1 e1 e3 e5
• PY,2 e2 e4 e8e9 e10
• Two edge-disjoint paths to Z
• PZ,1 e1 e3 e7e9e11
• PZ,2 e2 e4 e6
• Initialization
• eY,1 e1, eY,2 e2, eZ,1 e1, eZ,2 e2.
• fe10 1T, fe21 0T.
• fei0 0T, i3,4, ,11.

e1
e2
S
e3
e4
T
U
e7
e8
W
e9
e5
e6
X
e11
e10
Y
Z
66

• fe10 1T
• fe21 0T
• fe30 0T
• fe40 0T
• fe50 0T
• fe60 0T
• fe70 0T
• fe80 0T
• fe90 0T
• fe100 0T
• fe110 0T

• fe10 1T
• fe21 0T
• fe30 1T
• fe40 0T
• fe50 0T
• fe60 0T
• fe70 0T
• fe80 0T
• fe90 0T
• fe100 0T
• fe110 0T

• fe10 1T
• fe21 0T
• fe30 1T
• fe41 0T
• fe50 0T
• fe60 0T
• fe70 0T
• fe80 0T
• fe90 0T
• fe100 0T
• fe110 0T

• fe10 1T
• fe21 0T
• fe30 1T
• fe41 0T
• fe50 1T
• fe60 0T
• fe70 0T
• fe80 0T
• fe90 0T
• fe100 0T
• fe110 0T

• fe10 1T
• fe21 0T
• fe30 1T
• fe41 0T
• fe50 1T
• fe60 0T
• fe70 1T
• fe80 0T
• fe90 0T
• fe100 0T
• fe110 0T

• fe10 1T
• fe21 0T
• fe30 1T
• fe41 0T
• fe50 1T
• fe61 0T
• fe70 1T
• fe80 0T
• fe90 0T
• fe100 0T
• fe110 0T

• fe10 1T
• fe21 0T
• fe30 1T
• fe41 0T
• fe50 1T
• fe61 0T
• fe70 1T
• fe81 0T
• fe90 0T
• fe100 0T
• fe110 0T

• fe10 1T
• fe21 0T
• fe30 1T
• fe41 0T
• fe50 1T
• fe61 0T
• fe70 1T
• fe81 0T
• fe91 1T
• fe100 0T
• fe110 0T

• fe10 1T
• fe21 0T
• fe30 1T
• fe41 0T
• fe50 1T
• fe61 0T
• fe70 1T
• fe81 0T
• fe91 1T
• fe101 1T
• fe111 1T

eY,1eZ,1e1
eY,2eZ,2e2
e1
e2
S
e3
e4
eY,1eZ,1e3
eY,2eZ,2e4
eZ,1e3
eY,2e4
T
U
eY,2e8
e7
eZ,1e7
e8
W
e9
eY,2eZ,1e9
e5
e6
eY,1e5
eZ,2e6
X
eZ,1e11
eY,2e10
e11
e10
Y
Z
67
Construction of Linear Broadcast

• For a non-source node t, if maxflow(t) lt ?,
  install a new node t with ? incoming channels.
• Among the ? incoming channels, maxflow(t) of them
  come from t, while the others come from t.
• Construct a linear multicast on the expanded
  graph.

68
Construction of Linear Dispersion

• For a collection of non-source nodes T, install a
  new node uT.
• For each node t in T, install maxflow(t) channels
  from node t to node uT.
• Obviously, maxflow(uT) maxflow(T).
• Construct a linear broadcast on the expanded
  graph.
• The complexity can be exponential, even for a
  fixed ?, since there are 2N-1 subsets of nodes.

69
Generic Network Code

• Definition. An ?-dimensional linear network code
  on an acyclic network is said to be generic if
  for every collection of channels ? ? E, if
• ? min?,maxflow(?)
• then fe,e ? ? are linearly independent.
• The global encoding kernels of a generic network
  code is as linearly independent as possible.
• Generic network code ? Linear dispersion ? Linear
  broadcast ? Linear multicast

70
Generic Network Code (cont.)

• Can be constructed by deterministic algorithm.
• The field size needs to be at least
• The complexity is proportional to
• Random coding can be done for large F.

71
A Linear Dispersion not Generic
72
Network Coding vs Algebraic Coding
73
A Linear Multicast

• A message of k symbols from a base field F is
  generated at the source node s.
• A k-dimensional linear multicast has the
  following property A non-source node t can
  decode the message correctly if and only if
• maxflow(t) ? k.
• By the Max-flow bound, this is also a necessary
  condition for a node t to decode (for any given
  network code).
• Thus the tightness of the Max-flow bound is
  achieved by a linear multicast, which always
  exists for sufficiently large base fields.

74
An (n,k) Code with dmin d

• Consider a (n,k) classical block code with
  minimum distance d.
• Regard it as a network code on an
• combination network.
• Since the (n,k) code can correct d-1 erasures,
  all the nodes at the bottom can decode.

75
The Combination Network
s
n
n-d1
n-d1
76

• For the nodes at the bottom,
• maxflow(t) n-d1.
• By the Max-flow bound,
• k ? maxflow(t) n-d1
• or d ? n-k1, the Singleton bound.
• Therefore, the Singleton bound is a special case
  of the Max-flow bound for network coding.
• An MDS code is a classical block code that
  achieves tightness of the Singleton bound.
• Since a linear multicast achieves tightness of
  the Max-flow bound, it is formally a network
  generalization of an MDS code.

77
Applications of Random Network Coding in P2P
78
What is Peer-to-Peer (P2P)?

• Client-Server is the traditional architecture for
  content distribution in a computer network.
• P2P is the new architecture in which users who
  download the file also help disseminating it.
• Extremely efficiently for large-scale content
  distribution, i.e., when there are a lot of
  clients.
• P2P traffic occupies at least 70 of Internet
  bandwidth.
• BitTorrent is the most popular P2P system.

79
What is Avalanche?

• Avalanche is a Microsoft P2P prototype that uses
  random linear network coding.
• It is one of the first applications /
  implementations of network coding by Gkantsidis
  and Rodriguez 05.
• It has been further developed into Microsoft
  Secure Content Distribution (MSCD).
• A similar system has been built for UUSee.com.

80
How Avalanche Works?

• When the server or a client uploads to a
  neighbor, it transmits a random linear
  combination of the blocks it possesses. The
  linear coefficients are attached with the
  transmitted block.
• Each transmitted block is some linear combination
  of the original blocks of the seed file.
• Download is complete if enough linearly
  independent blocks have been received, and
  decoding can be done accordingly.

81
The Butterfly Network A Review
Synchronization here
82
What Exactly is Avalanche Doing?

• In Avalanche, there does not seem to be any need
  for synchronization.
• Is Avalanche doing the same kind of network
  coding we have been talking about?
• If not, what is it doing and is it optimal in any
  sense?

83
A Time-Parametrized Graph
84
Analysis of Avalanche(Y, NetCod 2007)

• The time-parametrized graph, not the physical
  network, is the graph to look at.
• By examining the maximum flows in this graph, the
  following questions can be answered
• When can a client receive the whole file?
• If the server and/or some clients leave the
  system, can the remaining clients recover the
  whole file?
• If some blocks are lost at some clients, how does
  it affect the recovery of the whole file?

85
Some Remarks

• Avalanche is not doing the usual kind of random
  network coding.
• It can be analyzed by the tools we are familiar
  with.
• Avalanche minimizes delay with respect to the
  given transmission schedule if computation is
  ignored.
• Extra computation is the price to pay.
• Avalanche provides the maximum possible
  robustness for the system.
• P2P is perhaps the most natural environment for
  applying random network coding because the subnet
  is formed on the fly.

86
Networks with Packet Loss A Toy Example

• One packet is sent on each channel per unit time.
• Packet loss rate 0.1 on each channel.
• By using a fountain code, information can be
  transmitted from A to C at rate (0.9)2 0.81.
• By using a fountain code with decoding at B,
  information can be transmitted from A to C at
  rate 0.9, but much delay is incurred.

87

• By using an Avalanche-type system, information
  can be transmitted from A to C at rate
• 0.9 max-flow
• from A to C.
• Why?

88
An Explanation
?
?
?
?
?
?
89
Networks with Packet Loss

• By using an Avalanche-type system, the max-flow
  from the source to the sink (amortized by the
  packet loss rate) can be achieved automatically,
  which is the fundamental limit.
• Virtually nothing needs to be done.
• Remark Avalanche has a clear advantage only
  there is no link-by-link feedback.

90
Single-Source Linear Network Coding

• Cyclic Networks

91
Unit-Delay Cyclic Networks

• We consider the following type of networks
• There may be directed cycles. It may not be
  possible to order the nodes in an
  upstream-to-downstream manner.
• There is only one information source generated at
  the node s.
• There is a unit-delay at each node to avoid
  information looping.

92
Formal Power Series

• In a formal power series (z-transform)
• c0 c1z c2z2
• z is a dummy variable should not be regarded as
  a real or complex number.
• The power j in zj represents discrete time.
• The power series should be regarded as nothing
  but a representation of the sequence c0, c1, c2,

93
Rational Power Series

• Important Properties
• The denominator has a constant term, so the
  function can be expanded into a power series by
  long division.
• If p(z) is not the zero polynomial, the inverse
  function
• exists.
• If p(z) contains the factor z, the inverse
  function is not a power series.

94
Algebraic Structures

• Fz - The ring of power series over F
• F?z? - The ring of rational power series over F
• F?z? is a subring of Fz.

95
Example

• If z is complex, for z lt 1, we can write
• If z is a dummy variable, we can always write

96

• In this sense, we say that 1-z is the reciprocal
  of 1zz2z3 and write
• 1zz2z3 can be obtained by dividing 1 by 1-z
  using long division.

97
Information Source

• At each discrete time t 0, the information
  source generates a row vector
• xt ( x1,t, x2,t, , x?,t ),
• where each symbol xi,t is in some finite field
  F.
• For 1 i ?, the pipeline of symbols xi,0,
  xi,1, xi,2, can be conveniently represented by
  a formal power series
• Xi(z) xi,0 xi,1 z xi,2 z2
• So the information source can be represented by
• x(z) ( x1(z) x2(z) x?(z) ).

98
Global Description

• Definition 20.6.
• An ?-dimensional convolutional network code on a
  unit-delay network consists of
• an element kd,e(z) ? F?z? for every adjacent pair
  of channels (d,e)
• an ?-dimensional column vector fe(z) ? F?z?? for
  every channel e
• such that
• fe(z) z ?d?In(T) kd,e(z) fd(z) for e ? Out(t).
• for the ? imaginary channels in In(S), the
  vectors fe(z) form the standard basis of the
  vector space F?.

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Global and Local Encoding Kernels

• fe(z) is the global encoding kernel for the
  channel e
• kd,e(z) is the local encoding kernel for the
  adjacent pair (d,e).
• The local encoding kernel at node t refers to the
  In(t)?Out(t) matrix
• KT(z) kd,e(z)d?In(T),e?Out(T).

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Input-Output Relation

• Let ye(z) be the power series representing the
  pipeline of symbols transmitted on channel e,
  given by x(z) fe(z).
• Then for e ? Out(t),
• ye(z) x(z) fe(z)
• x(z) z ?d?In(T) kd,e(z) fd(z)
• z ?d?In(T) kd,e(z) x(z) fd(z)
• z ?d?In(T) kd,e(z) yd(z)
• In other words, an output power series is the sum
  of the convolutions between an input power series
  with the corresponding local encoding kernels,
  with a unit delay.

101
Example
102
Existence of Convolutional Network Code

• The equation
• fe(z) z ?d?In(T) kd,e(z) fd(z)
• gives the relation between the global encoding
  kernels and the local encoding kernels.
• In time domain, it becomes the convolutional
  equation
• fe,t ?d?In(T) (?0?ultt kd,e,u fd,t?1?u)
• for all t gt 0, allowing the global encoding
  kernels to be computed from the local encoding
  kernels.

103
A Trellis Network
104
Existence of Convolutional Network Code(cont.)

• For any given local encoding kernels kd,e, fe
  can be calculated recursively. So, fe ? Fz?
  for all e.
• By definition, the vectors fe need to be in
  F?z? in order to be qualified as global encoding
  kernels. This is necessary for the purpose of
  decoding.
• The next theorem shows that this is indeed the
  case.

105
Existence of Convolutional Network Code(cont.)

• Theorem 20.9
• Let kd,e(z) ? F?z? be given for every adjacent
  pair of channels (d,e) on a unit-delay network.
  Then there exists a unique ?-dimensional
  convolutional network code with kd,e(z) as the
  local encoding kernel for every (d,e).

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Proof of Theorem 20.9

• Let fe(z) be the ??E matrix formed by putting
  all the global encoding kernels in juxtaposition
  (excluding the ? imaginary channels).
• Let kd,e(z) be the E?E matrix whose (d,e)th
  component is equal to kd,e(z) if (d,e) is an
  adjacent pair, and is equal to 0 otherwise.
• Let HS(z) be the ??E matrix formed by appending
  E -Out(s) columns of zeros to the local
  encoding kernel Ks(z).
• The columns of these matrices are assumed to
  indexed by some upstream-to-downstream order.

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• Then
• fe(z) z fe(z) kd,e(z) z HS(z).
• so that
• fe(z) z det(IN z kd,e(z))-1 HS(z) A(z),
• where A(z) is the adjoint matrix of
• IN z kd,e(z).
• Therefore the global encoding kernels can be
  expressed in closed-form in terms of the local
  encoding kernels (in frequency domain).
• Moreover, it is seen that fe(z) are in F?z?,
  i.e., they are rational. If not, decoding may
  not be possible.

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Convolutional Multicast

• Definition 20.10.
• An ?-dimensional convolutional network code
  qualifies as an ?-dimensional convolutional
  multicast if
• For every non-source node t with maxflow(t) ? ?,
  there exists an In(t)?? matrix Dt(z) over F?z?
  and a positive integer t such that
• fe(z)e?In(t)Dt(z) zt I?,
• where decoding delay t depends on the node t.
  The matrix Dt(z) is called the decoding kernel at
  node t.

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Lemma 20.12

• Let g(y1, y2, , ym) be a nonzero polynomial
  with coefficients in a field F. For any subset
  E of F, if E is greater than the degree of g
  in every yj, then there exist a1, a2, , an ? E
  such that
• g(a1, a2, , am) ? 0.
• Remark This is a generalization of Lemma 19.17.

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Existence of a Convolutional Multicast

• Theorem 20.13.
• There exists an ?-dimensional convolutional
  multicast over any base field F. Furthermore,
  the local encoding kernels of the convolutional
  multicast can be chosen in any sufficiently large
  subset F of F?z?.

111
Existence of a Convolutional Multicast(cont.)

• Theorem 20.13 says that the local encoding
  kernels can, for example, be chosen in
• a sufficiently large finite field F
• the set of all binary polynomials up to a
  sufficiently large degree.
• Proof of Theorem 20.13 is analogous to the proof
  of Theorem 19.20 (linear multicast).