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The Zero-Error Capacity of Compound Channels

The Zero-Error Capacity of Compound Channels. Jayanth Nayak & Kenneth Rose University of California, Santa Barbara. Outline. Preliminaries Problem definition Capacity Neither side has side-information Encoder has side-information Decoder has side-information

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The Zero-Error Capacity of Compound Channels

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

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

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

  6. Connected 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

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

  8. Connected 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

  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. Encoder Decoder The Compound Channel • Once chosen, DMC remains constant throughout the transmission. S Side Information = S

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

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

  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. Decoder Informed of S • Decoder can change with channel • Therefore

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

  17. Thank you

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