Achieving Capacity of Large Alphabet Discrete Memoryless Channels

It is observed that some communication situations fall into the large alphabet setting, in which the number of channel parameters and the number of channel uses are both large. To model such situations, we consider Discrete Memoryless Channels (DMCs) in which the input and output alphabet sizes increase along with the blocklength n. For known channels, we show that reliable communication at the sequence of channel capacities is possible if and only if the minimum between the square logarithms of the input and the output alphabet sizes grows sublinearly with n. For unknown channels with feedback, we show that universal channel coding can be supported if the input-output product alphabet size grows sublinearly with n.