On the ratio of prefix codes to all uniquely decodable codes with a given length distribution

TL;DR

This paper analyzes the ratio of prefix codes to all uniquely decodable codes with a given length distribution, providing bounds and asymptotic behavior, and finds exact values in specific cases.

Contribution

It establishes bounds for the ratio of prefix to all uniquely decodable codes and explores their asymptotic limits, including exact values for certain length distributions.

Findings

The infimum of the ratio is always greater than zero.

The ratio tends to 1 as alphabet size increases for fixed length.

The ratio tends to 0 as length increases for fixed alphabet size.

Abstract

We investigate the ratio $\rho_{n,L}$ of prefix codes to all uniquely decodable codes over an $n$-letter alphabet and with length distribution $L$. For any integers $n\geq 2$ and $m\geq 1$, we construct a lower bound and an upper bound for $\inf_L\rho_{n,L}$, the infimum taken over all sequences $L$ of length $m$ for which the set of uniquely decodable codes with length distribution $L$ is non-empty. As a result, we obtain that this infimum is always greater than zero. Moreover, for every $m\geq 1$ it tends to 1 when $n\to\infty$, and for every $n\geq 2$ it tends to 0 when $m\to\infty$. In the case $m=2$, we also obtain the exact value for this infimum.