MMI译码器对典型随机码和删除码都是渐近最优的
The MMI Decoder is Asymptotically Optimal for the Typical Random Code and for the Expurgated Code
摘要：
我们提供了两个结果有关的最大互信息（MMI）解码器的最优性。首先，我们证明了典型随机码在最佳最大似然（ML）译码器和MMI译码器下的误差指数相等。作为这一结果的推论，我们还表明，删除码的错误指数下ML和MMI解码器是相等的。这些结果加强了Csiszár和Körner的著名结果，根据该结果，这些解码器实现相等的随机编码误差指数，因为典型的随机码和删除码的误差指数严格高于随机编码误差指数，至少在低编码速率下。虽然MMI解码器在随机编码误差指数意义上的通用最优性很容易通过交换信道噪声上的期望值和系综上的期望值来证明，但是当涉及到典型的和删减的指数时，这种交换不再能够被执行。因此，MMI解码器的通用最优性的证明必须是完全不同的，并且证明是高度非平凡的。
英文摘要：
We provide two results concerning the optimality of the maximum mutual information (MMI) decoder. First, we prove that the error exponents of the typical random codes under the optimal maximum likelihood (ML) decoder and the MMI decoder are equal. As a corollary to this result, we also show that the error exponents of the expurgated codes under the ML and the MMI decoders are equal. These results strengthen the well known result due to Csiszár and Körner, according to which, these decoders achieve equal random coding error exponents, since the error exponents of the typical random code and the expurgated code are strictly higher than the random coding error exponents, at least at low coding rates. While the universal optimality of the MMI decoder, in the random-coding error exponent sense, is easily proven by commuting the expectation over the channel noise and the expectation over the ensemble, when it comes to typical and expurgated exponents, this commutation can no longer be carried out. Therefore, the proof of the universal optimality of the MMI decoder must be completely different and it turns out to be highly non-trivial.
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