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Uniformly reasonable source encoding is often practically impossible

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The decoding of efficiently encoded messages, from either probabilistic, nonprobabilistic, or unknown message sources, is shown to be often practically impossible. Iftau(S)is a running-time bound on the computational effort of a decoderPsiaccepting a codewordPfor messageS, andgamma[K_{Psi}(S)]is an upper bound to acceptable codeword lengthmid P midwhen the shortest codeword forShas lengthK_{Psi}(S), then for many message sourcesmathcal{M}there exist messagesS in mathcal{M}such that: 1) if the encoder satisfiesgamma, then the decoder violatestau; 2) if the decoder satisfiestau, then the encoder violatesgamma. These conclusions remain valid even when we allow the decoder to reconstruct only an approximationS primein a neighborhooddelta(S)ofS. The compatibility of these results with those of information theory rests upon the fact that we are inquiring into the detailed properties of coding systems for individual messages and not into the ensemble average properties.

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Information Theory, IEEE Transactions on  (Volume:21 ,  Issue: 4 )