On the Maximum Size of a Prefix Code

A prefix code minimal with respect to a bitstring
$x$
is a prefix code where
$x$
is a concatenation of its codewords and it is minimal with respect to this property. What is the maximum size
$M(n)$
among all minimal codes over all bitstrings of length
$n?$
In this paper we determine the value of
$M(n)$
for all natural numbers
$n$
, discuss its computational complexity, relation to the Lambert function, provide tight upper bounds, and describe how the value of
$M(n)$
enables one to construct efficiently a Huffman code in the case of uniform probability distribution of the codewords.