Lossless and lossy source compression with near-uniform output: Is common randomness always required?

It is known that a sub-linear rate of source-independent random seed (common randomness) can enable the construction of lossless compression codes whose output is nearly uniform under the variational distance (Chou-Bloch-ISIT'13). This work uses finite-blocklength techniques to present an alternate proof that for near-uniform lossless compression, the seed length has to grow strictly larger than √n, where n represents the blocklength of the lossless compression code. In the lossy setting, we show the surprising result that a seed is not required to make the encoder output nearly uniform.