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

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 φ_n(Xⁿ) of length l(φ_n(Xⁿ)). Letting ε_n denote the decoding error probability, we consider the following two criteria on FV codes: i) ε_n = 0 for all n ≥ 1 and ii) lim sup_n→∞ε_n ≤ 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(φ_n(Xⁿ)) - 1/nlog₂ 1/PXⁿ(Xⁿ) → 0 in probability as n → ∞ under a certain condition, where P_Xⁿ 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(φ_n(Xⁿ)) of an arbitrary FV code achieving the minimum rate in a certain sense has a property similar to the lossless case.