On the achievable redundancy rate of fixed length source code for general sources

This paper is concerned with the redundancy rate of fixed length source code for a general source with a countably infinite alphabet. We evaluate the minimum achievable redundancy rate R of fixed-to-fixed length (FF) and variable-to-fixed length (VF) codes with two definitions of redundancy rates, which are (i) the difference between the coding rate and the spectral sup-entropy rate and (ii) the difference between the coding rate and the self information rate. First we show that, when we restrict the fixed-length code class within the class of FF codes, R with definition (i) is zero, but R with definition (ii) can be positive. Next we show that, by taking the VF codes into account, R with definition (ii) can be decreased to zero.