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Identification without randomization

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2 Author(s)
Ahlswede, R. ; Fak. fur Math., Bielefeld Univ., Germany ; Ning Cai

In the theory of identification via noisy channels randomization in the encoding has a dramatic effect on the optimal code size, namely, it grows double-exponentially in the blocklength, whereas in the theory of transmission it has the familiar exponential growth. We consider now instead of the discrete memoryless channel (DMC) more robust channels such as the familiar compound (CC) and arbitrarily varying channels (AVC). They can be viewed as models for jamming situations. We make the pessimistic assumption that the jammer knows the input sequence before he acts. This forces communicators to use the maximal error concept and also makes randomization in the encoding superfluous. Now, for a DMC W by a simple observation, made by Ahlswede and Dueck (1989), in the absence of randomization the identification capacity, say CNRI(W), equals the logarithm of the number of different row-vectors in W. We generalize this to compound channels. A formidable problem arises if the DMC W is replaced by the AVC W. In fact, for 0-1-matrices only in W we are-exactly as for transmission-led to the equivalent zero-error-capacity of Shannon. But for general W the identification capacity CNRI(W) is quite different from the transmission capacity C(W). An observation is that the separation codes of Ahlswede (1989) are also relevant here. We present a lower bound on C NRI(W). It implies for instance for W={(0 11 0), (δ (1-δ)(1 0))}, δ∈(0, ½) that CNRI(W)=1, which is obviously tight. It exceeds C(W), which is known to exceed 1-h(δ), where h is the binary entropy function. We observe that a separation code, with worst case average list size L¯ (which we call an NRA code) can be partitioned into L¯2ne transmission codes. This gives a nonsingle-letter characterization of the capacity of AVC with maximal probability of error in terms of the capacity of codes with list decoding. We also prove that randomization in the decoding does not increase CI(W) and CNRI(W). Finally, we draw attention to related work on source coding

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Information Theory, IEEE Transactions on  (Volume:45 ,  Issue: 7 )