On the Maximum Size of a Prefix Code

Abstract

A prefix code P 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.