Network Coding Theory:
Consolidation and Extensions
Raymond Yeung
Joint work with
Bob Li, Ning Cai and Zhen Zhan
Outline
 Single-Source Network Coding



Global and Local Descriptions of a Network
Code
Linear Multicast, Broadcast, and Dispersion
Static codes
 Multi-Source Network Coding

Fundamental Limits of Linear Codes
Based on an upcoming paper to appear in
Foundation and Trends in Communications and
Information Theory (Editor: Sergio Verdu).
Single-Source Network Coding
 Network is acyclic.
 The message x, a -dimensional row vector
in F, is generated at the source node.
 A symbol in F can be sent on each channel.
Global Description
 The symbol sent on channel e is a function of
the message, called the global encoding
mapping for channel e.
 For any node v, the global encoding
mappings have to satisfy the local constraints,
i.e., the local encoding mapping for every
node v is well defined.
A Globally Linear Network Code
 A code is globally linear if all the global encoding
mappings are linear (and all the local constraints
satisfied).
 A globally linear code is the most general linear code
that can possibly be defined.
 The global encoding mapping for channel e is
characterized by a column vector fe, s.t. the symbol
sent on e is x fe.
 It can be proved that if a code is globally linear, then
it is also locally linearly, i.e., all local encoding
mappings are linear.
Global Description vs Local Description
 Since the local encoding mapping at a node v
is linear, it follows that
for any e Out(v), fe is a linear combination of
fe’, e’  In(v).
 Global description (Li-Yeung-Cai).
 These linear combination forms the local
encoding kernel.
 Local description (Koetter-Medard)
Global Description = Local Description
 The global description and the local
description are the two sides of a coin:


They are equivalent.
Both can describe the most general form of a
(block) linear network code!
Generic Network Code
 Definition (LYC)
A linear network code is said to be generic if:
For every set of channels {e1, e2, … , en},
where n   and ej  Out(vj), the vectors fe1,
fe2, … , fen are linearly independent provided
that
{fd: d  In(vj)}  {fek: k  j} for 1  j  n.
 The idea: Whenever a collection of vectors
can possibly be linear independent, they are.
Special Cases of a Generic
Network Code
Generic network code
 Linear dispersion
 Linear Broadcast
 Linear Multicast
Each notion is strictly weaker than
the previous notion!
Linear Multicast
 For each node v, if maxflow(v)  , then the
message x can be recovered.
Linear Broadcast
 For every node v,


If maxflow(v)  , the message x can be
received.
If maxflow(v) < , maxflow(v) dimensions of
the message x can be recovered.
 Linear Broadcast  Linear Multicast
Linear Dispersion
 For every collection of nodes P,


If maxflow(P)  , the message x can be received.
If maxflow(P) < , maxflow(P) dimensions of the
message x can be recovered.
 Linear Dispersion  Linear Broadcast
 Linear Mulicast (Generic network code implies all)
 For a linear dispersion, a new comer who wants to
receive the message x can do so by accessing a
collection of nodes P such that maxflow(P)  , where
each individual node u in P may have maxflow(u) < .
Code Constructions
 A generic network code exists for all
sufficiently large F and can be constructed by
the LYC algorithm.
 A linear dispersion, a linear broadcast, and a
linear multicast can potentially be constructed
with decreasing complexity since they satisfy
a set of properties of decreasing strength.
 In particular, a polynomial time algorithm for
constructing a linear multicast has been
reported independently by Sanders et al. and
Jaggi et al.
Static Codes
 Static linear multicast was introduced by KM
which finds applications in robust network
multicast.
 Static versions of linear broadcast and linear
dispersion can be defined accordingly.
 The LYC algorithm can be modified for
constructing a static generic network code.
 This means that the static versions of a linear
dispersion, a linear broadcast, and a linear
multicast can all be constructed.
Multi-Source Network Coding
 A network is given.
 Independent information sources of rates
 = (1, 2, …, S)
are generated at possibly different nodes, and each
source is to be multicast to a specific sets of nodes.
 The set of all achievable rates is called the
achievable information rate region R.
 If all the sources are multicast to the same set of
nodes, then it reduces to a single-source network
coding problem, otherwise it does not.
 A multi-source network coding problem
cannot be decomposed into single-source
network coding problems even when all the
information sources are generated at the
same node (Yeung 95).
 Special multi-source network coding
problems have been shown to be
decomposable (Roche, Hau, Yeung, Zhang
95-99).
An Example of Indecomposability
(with Wireless Application)
Independent sources need to be coded jointly
b1
b1
b2
b2
b1+b2
b1
b2
Characterization of the Information
Rate region R
 Inner and outer bounds on R acyclic networks
can be expressed in term of the region of all
entropy functions of random variables (Yeung
97, Yeung-Zhang 99, Song et al. 03).
 A computable outer bound on R, called RLP,
has also been obtained.
 Only existence proofs by random coding are
available  no code construction.
The region Γ*
 Let Γ* be the set of all entropy functions of a
collection of random variables labeled by the
information sources and the channels.
Outer Bound Rout
If an information rate tuple  is achievable, then there
exists h  closure(Γ*) which satisfies a set of
constraints denoted by C which specifies
the independence of the information sources
2. the rate tuple
3. local constraints of the code
4. the channel capacity constraints
5. the multicast requirements.
C is a collection of hyperplanes in the Eucledian space.
1.
Linear Codes for Multiple Sources
 The global description for a linear network
code can be generalized to multiple sources.
 Each channel is characterized by a column
vector of an appropriate dimension.
 The existence of a linear code is nothing but
the existence of a collections of vectors
satisfying the set of constraints C.
The Region *
 Let * be the set of all rank functions for a
collection of -dimensional column vectors
labeled by the information sources and the
channels over some finite field F, where
  1.
Linear Codes vs Nonlinear Codes
Linear codes  Rlinear
An information rate tuple  is linearly
achievable iff there exists h  closure(*)
which satisfies the set of constraints C.
Note: Rlinear includes all rate tuples that are inferior to some rate tuples
achievable by mixing linear codes.
Nonlinear codes  outer bound Rout
If an information rate tuple  is achievable, then
there exists h  closure(Γ*) which satisfies
the set of constraints C.
Similarity between Rank and Entropy
 The rank function satisfies
1. 0  rank(A).
2. rank(A)  rank(B) if A  B.
3. rank(A) + rank(B)  rank(AB) + rank(A B).
4. rank(A)  |A|.
 The entropy function in general satisfies
1. 0  H(A).
2. H(A)  H (B) if A  B.
3. H(A) + H (B)  H (AB) + H (A B).
1 - 3 are called the polymatroidal axioms.
The Bridge from Rank to Entropy
Theorem 1:
Let F be a finite field, Y be an -dimensional
random row vector that distributes uniformly
on F, and A be an   l matrix. Let Z = Y·A.
Then H(Z) = rank(A) log |F|.
Using this theorem, it can be shown that
*  Γ*.
A Gap between * and Γ*
 In addition to the polymatroidal axioms, the rank
function also satisfies the Ingleton inequality:
r(A13)+ r(A14)+ r(A23)+ r(A24)+ r(A34)
 r(A3)+ r(A4)+ r(A12)+ r(A134)+ r(A234)
 The Ingleton inequality is satisfied by algebraic
structures as general as Abelian groups.
 The corresponding inequality is not satisfied by the
entropy function (Zhang-Yeung 99), so there is a gap
between * and Γ*.
 This gap between * and Γ* suggests that nonlinear
codes may actually perform better for some multisource problems.
Vector Linear Codes
 Vector Linear Codes (Riis, Lehman2, Medard, Effros,
Ho, Karger, Koetter)


It can be regarded as a linear code over a network
obtained by expanding all the capacities by an integer
factor.
It has been shown that some multi-source problems do
not have linear solutions but have vector linear
solutions.
 Question 1: Are these vector linear solutions better
than all mixtures of linear solutions?
Question 2: Do these vector linear solutions exceed
the Ingleton inequality? (If so, the answer to Q1 is
yes.)
Codes Beyond Fields
 Dougherty, Frieling and Zeger have recently
shown that there exist a multi-source problem
that has no linear solution even in the more
general algebraic context of modules, which
includes all finite rings and Abelian groups.
 Question 1: Is the nonlinear solution given by
DFZ better than all mixtures of linear
solutions?
Question 2: Does the nonlinear solution given
by DFZ exceed the Ingleton inequality? (If so,
the answer to Q1 is yes.)
Ingleton Inequality Classification
 Codes abide by the Ingleton inequality
 Linear codes, module codes
 Codes not necessarily abide by the Ingleton
inequality

Vector linear codes (abide by the Ingleton
inequality in an extended space)
 Codes not abide by the Ingleton inequality
 Non-Abelian group codes are asymptotically
as good as all nonlinear codes (Chan,
submitted to ISIT 2005).
Thank You