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Asymptotic properties on codeword lengths of an optimal FV code for general sources

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2 Author(s)
H. Koga ; Graduate Sch. of Syst. & Inf. Eng., Univ. of Tsukuba, Ibaraki, Japan ; H. Yamamoto

This correspondence is concerned with asymptotic properties on the codeword length of a fixed-to-variable length code (FV code) for a general source {Xn}n=1 with a finite or countably infinite alphabet. Suppose that for each n ≥ 1 Xn is encoded to a binary codeword φn(Xn) of length l(φn(Xn)). Letting εn denote the decoding error probability, we consider the following two criteria on FV codes: i) εn = 0 for all n ≥ 1 and ii) lim supn→∞εn ≤ ε for an arbitrarily given ε ∈ [0,1). Under criterion i), we show that, if Xn is encoded by an arbitrary prefix-free FV code asymptotically achieving the entropy, 1/nl(φn(Xn)) - 1/nlog2 1/PXn(Xn) → 0 in probability as n → ∞ under a certain condition, where PXn denotes the probability distribution of Xn. Under criterion ii), we first determine the minimum rate achieved by FV codes. Next, we show that 1/nl(φn(Xn)) of an arbitrary FV code achieving the minimum rate in a certain sense has a property similar to the lossless case.

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IEEE Transactions on Information Theory  (Volume:51 ,  Issue: 4 )