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

TL;DR

This paper investigates the structure and relationships of uniquely decodable codes with fixed length sequences, providing estimations and characterizations for when these codes coincide with prefix codes and finite delay codes.

Contribution

It offers new estimations for the ratio of uniquely decodable codes to prefix codes and characterizes sequences where these classes are equal.

Findings

Derived estimation for |UD_n(L)|/|PR_n(L)|

Characterized sequences where PR_n(L)=UD_n(L)

Characterized sequences where FD_n(L)=UD_n(L)

Abstract

For every natural number $n\geq 2$ and every finite sequence $L$ of natural numbers, we consider the set $UD_n(L)$ 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 $PR_n(L)$ and $FD_n(L)$ consisting of, respectively, the prefix codes and the codes with finite delay. We derive the estimation for the quotient $|UD_n(L)|/|PR_n(L)|$, which allows to characterize those sequences $L$ for which the equality $PR_n(L)=UD_n(L)$ holds. We also characterize those sequences $L$ for which the equality $FD_n(L)=UD_n(L)$ holds.