1. The Zero-Error Capacity of Compound Channels Jayanth Nayak & Kenneth Rose University of California, Santa Barbara

2. 2 Outline Preliminaries Problem definition Capacity –Neither side has side-information –Encoder has side-information –Decoder has side-information Comparison with vanishing-error case

3. 3 Zero-Error vs. Vanishing-Error Vanishing-error Case – tends to zero as block length tends to infinity. Zero-error case – zero at all block lengths –Tools: graph theory and combinatorics

4. 4 Discrete Memoryless Channel Characteristic graph of the channel Example Channel transition probability 1 2 3   a b d e c 1 2 3 GXGX Undirected

5. 5 Channel Code Code  symbols pair-wise connected = clique Example (cont.) –Scalar code: 1-use capacity = – is the clique number bits 1 2 3 GXGX

6. 6 Channel Code uses of the channel –Fan-out set is Cartesian product of fan-out sets at each coordinate – = size of largest pairwise connected set Connected

7. 7 Channel Code Zero error capacity (Shannon ’56) Depends only on characteristic graph – Shannon capacity of a graph bits/channel use

8. 8 Other Graph Capacities Shannon capacity of a set of graphs: –Asymptotic size of set of sequences connected with respect to every graph from –Defined by Cohen et al. ‘88 Directed graphs Connected

9. 9 Other Graph Capacities Sperner capacity of a graph – largest pairwise connected set : connected with respect to every graph Sperner capacities defined by Gargano et al. ’94 –Motivated by extremal set theory

10. 10 The Compound Channel Once chosen, DMC remains constant throughout the transmission. EncoderDecoder Side Information = S S

11. 11 Capacity Conventional Case: –Dobrushin ‘59, Wolfowitz ‘60, Breiman ‘59 Zero-Error Case: –Cohen et al. ‘88, Gargano et al. ‘94 –Capacity expression accurate only when decoder is informed.

12. 12 Characteristic Set of Graphs Fan-out set – –Vertex set – –Directed graph

13. 13 Capacity of Compound Channel Every code is a pairwise connected set with respect to and vice versa

14. 14 Encoder Informed of S If >0, encoder sends side- information to decoder. Needs constant number of channel uses. = 0 case: – contains an edge-free graph

15. 15 Decoder Informed of S Decoder can change with channel – Therefore –

16. 16 Conventional vs. Zero-Error Conventional Capacities Zero-Error Capacities Reason: Decoder can identify channel with high probability

17. Thank you