Consolidation and Extensions of Network Coding Theory

1. Network Coding Theory:Consolidation and Extensions Raymond Yeung Joint work with Bob Li, Ning Cai and Zhen Zhan

2. 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).

3. 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.

4. 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.

5. 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.

6. 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)

7. 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!

8. 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.

9. 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!

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

11. 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

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

13. 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.

14. 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.

15. 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 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.

16. 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).

17. An Example of Indecomposability(with Wireless Application) Independent sources need to be coded jointly b1 b2 b1 b2 b1+b2 b1 b2

18. 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.

19. 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.

20. Outer Bound 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.

21. 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.

22. 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.

23. 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.

24. Similarity between Rank and Entropy • 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.

25. 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*.

26. 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 multi-source problems.

27. 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.)

28. 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.)

29. 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).

30. Thank You