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

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 inf_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.