Analogy and duality between random channel coding and lossy source coding

Here we write in a unified fashion (using “R(P,Q,D)” [1]) the random coding exponents in channel coding and lossy source coding. We derive their explicit forms and show, that, for a given random codebook distribution Q, the channel decoding error exponent can be viewed as an encoding success exponent in lossy source coding, and the channel correct-decoding exponent can be viewed as an encoding failure exponent in lossy source coding. We then extend the channel exponents to arbitrary D, which corresponds for D > 0 to erasure decoding and for D < 0 to list decoding. For comparison, we also derive the exact random coding exponent for Forney’s optimum tradeoff decoder [2]. In the case of source coding, we assume discrete memoryless sources with a finite alphabet X and a finite reproduction alphabet X̂ . In the case of channel coding, we assume discrete memoryless channels with finite input and output alphabets X and Y , such that for any (x, y) ∈ X ×Y the channel probability is positive P (y | x) > 0. For simplicity, let R denote an exponential size of a random codebook, such that there exist block lengths n for which e is integer. We assume the size of the codebook M = e for source coding, and M = e + 1 for channel coding. Let Q denote the (i.i.d.) distribution, according to which the codebook is generated. We use also the definition: