The Birthday Problem and Zero-Error List Codes

TL;DR

This paper explores the connection between zero-error list decoding in information theory and the birthday problem, providing an information-theoretic framework to analyze birthday coincidences across various distributions.

Contribution

It introduces an information-theoretic formulation of the birthday problem and analyzes the performance of random codebooks under zero-error list-decoding constraints.

Findings

Exact asymptotic behavior of birthday probabilities for all distributions

Application of probabilistic tools to zero-error information theory

New insights into codebook performance under zero-error constraints

Abstract

As an attempt to bridge the gap between the probabilistic world of classical information theory and the combinatorial world of zero-error information theory, this paper studies the performance of randomly generated codebooks over discrete memoryless channels under a zero-error list-decoding constraint. This study allows the application of tools from one area to the other. Furthermore, it leads to an information-theoretic formulation of the birthday problem, which is concerned with the probability that in a given population, a fixed number of people have the same birthday. Due to the lack of a closed-form expression for this probability when the distribution of birthdays is not uniform, the resulting expression is not simple to analyze; in the information-theoretic formulation, however, the asymptotic behavior of this probability can be characterized exactly for all distributions.