On the cost of universality of block codes for individual sequences

Consider a rate R block code of size k that achieves minimum distortion for a given individual input sequence. Such a code obviously depends on the input sequence. The author seek a universal (sequence-independent) rate R´ block code of size l (l>k), which guarantees a distortion not higher than that of the above optimal sequence-dependent block code. They ask the question of the cost of universality: what is the best achievable rate R´ as a function of the block length l of the universal code such that the above task could be fulfilled? They provide achievable bounds that characterize this optimal rate. Dual results on the cost of universality are shown for universal channel coding and distortionless joint source-channel coding