About Optimal Prefix Codes over Countably Infinite Alphabets: Probabilistic Intervals for the Codeword Lengths Assignment

TL;DR

This paper establishes probabilistic intervals for optimal codeword lengths in infinite alphabet sources and provides a criterion to identify distributions with linearly increasing code lengths, improving verification efficiency.

Contribution

It introduces probabilistic intervals for code lengths and a less demanding criterion for specific distribution patterns in infinite alphabet coding.

Findings

Existence of probability intervals for each code length k.

Criterion for distributions with linearly increasing code lengths.

Improved verification process compared to previous methods.

Abstract

For the discrete memoryless sources with a countably infinite alphabet, we prove that for any positive integer $k$, there exists a corresponding probability interval such that if the largest symbol probability $p_{1}$ falls in this interval, the optimal code length for the symbol equals $k$. Furthermore, for infinite sources, we provide a criterion to determine probability distributions whose optimal code length assignment follows the pattern $l^{best}_{i}=i$, for $i\ge 1$. Compared with the existing conclusion for anti-uniform sources, the proposed criterion requires less information for verification.