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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

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### Network Coding Theory:Consolidation and Extensions

Raymond Yeung

Joint work with

Bob Li, Ning Cai and Zhen Zhan

• 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).

• 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.

• 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 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.

• 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)

• 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!

• 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 Multicast

Each notion is strictly weaker than the previous notion!

• For each node v, if maxflow(v)  , then the message x can be recovered.

• 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

• 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 thatmaxflow(P)  , where each individual node u in P may have maxflow(u) < .

• 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 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.

• 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 regionR.

• 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 into single-source network coding problems (with Wireless Application)

Independent sources need to be coded jointly

b1

b2

b1

b2

b1+b2

b1

b2

Characterization of the Information Rate region into single-source network coding problems 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 into single-source network coding problems 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 into single-source network coding problems Rout

If an information rate tuple  is achievable, then there exists h  closure(Γ*) which satisfies a set of constraints denoted by Cwhich specifies

• the independence of the information sources

• the rate tuple

• local constraints of the code

• the channel capacity constraints

• the multicast requirements.

C is a collection of hyperplanes in the Eucledian space.

Linear Codes for Multiple Sources into single-source network coding problems

• 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 into single-source network coding problems *

• 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 into single-source network coding problems

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 into single-source network coding problems

• The rank function satisfies

• 0  rank(A).

• rank(A)  rank(B) if A  B.

• rank(A) + rank(B)  rank(AB) + rank(AB).

• rank(A)  |A|.

• The entropy function in general satisfies

• 0  H(A).

• H(A)  H (B) if A  B.

• H(A) + H (B)  H (AB) + H (AB).

1 - 3 are called the polymatroidal axioms.

The Bridge from Rank to Entropy into single-source network coding problems

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 into single-source network coding problems * 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 multi-source problems.

Vector Linear Codes into single-source network coding problems

• 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 into single-source network coding problems

• 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 into single-source network coding problems

• 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 into single-source network coding problems