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- W2564657638 abstract "One-to-one codes are one shot codes that assign a distinct codeword to source symbols and are not necessarily prefix codes (more generally, uniquely decodable). For example, such codes arise when there exists an end of message channel symbol. Interestingly, as Wyner proved in 1972, for such codes the average code length can be smaller than the source entropy. By how much? We call this difference the anti-redundancy. Various authors over the years have shown that the anti-redundancy can be as big as minus the logarithm of the source entropy. However, to the best of our knowledge precise estimates do not exist. In this note, we consider a block code of length n generated by a binary memoryless source, and prove that the average anti-redundancy is -(1/2)log <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sub> n + C + F(n) + o(1) where C is a constant and either F(n) = O if log <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sub> (1 - p)/p is irrational (where p is the probability of generating a 0) or otherwise F(ri) is a fluctuating function as the code length increases. This relatively simple finding requires a combination of quite sophisticated analytic tools such as precise evaluation of Bernoulli sums, the saddle point method, and theory of distribution of sequences modulo 1" @default.
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- W2564657638 date "2005-01-01" @default.
- W2564657638 modified "2023-09-23" @default.
- W2564657638 title "One-to-one code and its anti-redundancy" @default.
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