Abstract
We show that the maximin average redundancy in pattern coding is eventually larger than 1.84 (n/log n)1/3 for messages of length n. This improves recent results on pattern redundancy, although it does not fill the gap between known lower- and upper-bounds. The pattern of a string is obtained by replacing each symbol by the index of its first occurrence. The problem of pattern coding is of interest because strongly universal codes have been proved to exist for patterns while universal message coding is impossible for memoryless sources on an infinite alphabet. The proof uses fine combinatorial results on partitions with small summands.
Highlights
Let P be a stationary source on an alphabet A, both known by the coder and the decoder
For a positive integer n, we denote by X1n the vector of the n first components of X and by Pn the distribution of X1n on An
We denote the logarithm with base 2 by log and the natural logarithm by ln
Summary
Let P be a stationary source on an alphabet A, both known by the coder and the decoder. If P is only known to be an element Pθ of some class C = {Pθ : θ ∈ Θ}, universal coding consists in finding a single code, or equivalently a single sequence of coding distributions (qn )n , approaching the entropy rate for all sources Pθ ∈ C at the same time. Such versatility has a price: for any given source. If the alphabet A is infinite, Kieffer [7] showed that no universal coding is possible even for the class of memoryless processes
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