On the Maximum Size of a Prefix Code

In this paper, we investigate the maximum size of a minimal dictionary of a binary prefix-code string. We develop exact formulas for the maximum number of codewords of a minimal dictionary, which belongs to a binary string of some length. Further, we elaborate on the computational complexity of our approach and its relation to the Lambert function. We also present a way, how this information enables us to efficiently construct a Huffman code in the case of uniform probability distribution of codewords.

The paper is of mathematical nature, i.e. all the methodology used in the paper is based on mathematical proofs.