On the set of uniquely decodable codes with a given sequence of code word lengths

For every natural number n ≥ 2 and every finite sequence L of natural numbers, we consider the set UDn(L) of all uniquely decodable codes over an n-letter alphabet with the sequence L as the sequence of code word lengths, as well as its subsets PRn(L) and FDn(L) consisting of, respectively, the prefix codes and the codes with finite delay. We derive the estimation for the quotient |UDn(L)|/|PRn(L)|, which allows to characterize those sequences L for which the equality PRn(L) = UDn(L) holds. We also characterize those sequences L for which the equality FDn(L) = UDn(L) holds.