Universal codes in the shared-randomness model for channels with general distortion capabilities
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We put forth new models for universal channel coding. Unlike standard codes which are designed for a specific type of channel, our most general universal code makes communication resilient on every channel, provided the noise level is below the tolerated bound, where the noise level t of a channel is the logarithm of its ambiguity (the maximum number of strings that can be distorted into a given one). The other more restricted universal codes still work for large classes of natural channels. In a universal code, encoding is channel-independent, but the decoding function knows the type of channel. We allow the encoding and the decoding functions to share randomness, which is unavailable to the channel. There are two scenarios for the type of attack that a channel can perform. In the oblivious scenario, codewords belong to an additive group and the channel distorts a codeword by adding a vector from a fixed set. The selection is based on the message and the encoding function, but not on the codeword. In the Hamming scenario, the channel knows the codeword and is fully adversarial. For a universal code, there are two parameters of interest: the rate, which is the ratio between the message length k and the codeword length n, and the number of shared random bits. We show the existence in both scenarios of universal codes with rate 1-t/n - o(1), which is optimal modulo the o(1) term. The number of shared random bits is O(log n) in the oblivious scenario, and O(n) in the Hamming scenario, which, for typical values of the noise level, we show to be optimal, modulo the constant hidden in the O() notation. In both scenarios, the universal encoding is done in time polynomial in n, but the channel-dependent decoding procedures are in general not efficient. For some weaker classes of channels we construct universal codes with polynomial-time encoding and decoding.
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