On the ratio of prefix codes to all uniquely decodable codes with a given length distribution

04/07/2018

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by   Adam Woryna, et al.

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We investigate the ratio ρ_n,L of prefix codes to all uniquely decodable codes over an n-letter alphabet and with length distribution L. For any integers n≥ 2 and m≥ 1, we construct a lower bound and an upper bound for _Lρ_n,L, the infimum taken over all sequences L of length m for which the set of uniquely decodable codes with length distribution L is non-empty. As a result, we obtain that this infimum is always greater than zero. Moreover, for every m≥ 1 it tends to 1 when n→∞, and for every n≥ 2 it tends to 0 when m→∞. In the case m=2, we also obtain the exact value for this infimum.