On α-decodability and α-likelihood decoder

Generalizing the maximum and the average error criteria for channel coding, we introduce the α-decodability, defined as the α-norm of the probabilities of correctly decoding the messages. Several aspects, such as the exponent, the existence of a strong Fano’s inequality, and the achievability of the channel capacity by random coding are investigated, and it is revealed that α “ 0 (corresponding to the geometric average of the probabilities) emerges as the critical value for several properties. In the same vein of interpolating the maximum and the average, we also revisit the α-likelihood decoder, which outputs a message with probability proportional to the its likelihood to the power α. For any code, up to a factor of 2, we show that the average error probability of the 1-likelihood decoder is optimal, and that the error probability of the 0-likelihood decoder coincides with the zero undetected error probability. This provides a unified approach to the (conventional) channel coding and the zero undetected error coding, and strengthens previous results on the error exponents of the likelihood decoder with simpler proofs. We also establish a connection between the parameter α in the likelihood decoder and the parameter ρ in Gallager’s random coding exponent.