1. Distributed zero-error network coding Tracey Ho Michelle Effros Sidharth Jaggi

2. s t1t1 t2t2 [ACLY00] [JCJ03], [SET03] No error Centralized design [HKMKE03], [JCJ03] Prob(error) ~2 -n€ Decentralized design...... No error Decentralized Design Complexity (Open CS problems) Decentralized zero-error

3. Convolutional network codes b1b1 b2b2 F 2 (z)-linear network [ACLY00],[KM02],… Source:- Generates a bit-stream Every node:- Perform linear combinations over field F 2 (z) …,[JHEM04],… Given an algebraic network code over field of size q, exists a convolutional network code with degree-log(q) polynomials degree-m polynomials 2 m =q

4. Decentralized Information about network structure/code percolates along network links zero-error No error! Guaranteeing linear independence of information in each cut-set… … is impossible! (Proof by example – exercise for audience) Need some global information about network.

5. Decentralized zero-error Each node has unique ID # i Toy example (C=2 (~[FS04])) 1.One linearly independent incoming vector Replicate on all outgoing links 2. Two linearly independent incoming vectors Maintain “MDS” property i

6. Decentralized zero-error Really cool! Lexicographic ordering (Cantor-diagonal assignment) m = polylog(i,j)

7. Decentralized zero-error General C, existence proof… Complete graph with |V| nodes and C links from v i to v j, for all (i,j) # subgraphs with mincut C ≤ # subgraphs = 2 C|V|(|V|-1) [KM02] Can design ONE network code which works for MANY networks… … Field size proportional to 2 C|V|(|V|-1) … m proportional to C|V|(|V|-1) (exponential increase, non-constructive :-/)

8. Decentralized zero-error General C, “construction” [HMSEK04] Network code works [KM02] “Line graph matrix” f(i,j) distinct for each (i,j) pair (Cantor-diagonal assignment) Each z term has distinct exponent Σ (i,j) in term 2 f(i,j) distinct in each term DOUBLE exponential field-size :-( Oh well… next paper… ;-)

9. Summary/Where now? C=2 toy example General C, exponentially larger fields required.  High computational complexity OR  High (double-exp) implementation complexity How much global info needed for tractable zero-error design?