List Reconstruction Problem with List Size Two

2026-05-22

AI Summary

This work investigates the maximum asymptotic rate of the intersection of three Hamming balls of radius \( r = \alpha n \) whose centers are pairwise separated by distance \( k = \beta n \), in the limit as the code length \( n \to \infty \). Motivated by applications in list-recoverable coding and associative memory, the study combines tools from combinatorics, information theory, and asymptotic analysis to provide the first explicit characterization of this three-ball intersection rate as a function of the parameters \( \alpha \) and \( \beta \). In contrast to the well-understood two-ball case, the results uncover distinctive structural properties inherent to multi-ball intersections. Numerical experiments corroborate the theoretical predictions, demonstrating both their accuracy and the qualitative differences from the classical pairwise setting.

Abstract

The problem of computing the cardinality of the intersection of multiple balls in the Hamming space has attracted a lot of attention recently due to their applications in the list reconstruction problem and information retrieval in Associative Memories. In previous work, most of the results are for the cases where the radii of each ball, $r$ and the distance between the centers of these balls, $k$ are fixed when the length $n$ of each codeword tend to infinity. In this work, we focus on the case where $r = αn$ and $k=βn$ for some constants $α$ and $β$ and compute the maximum asymptotic rate of the cardinality of the intersection of three balls. We provide the maximum asymptotic rate as a function of two parameters $α$ and $β$. We also provide numerical results and compare these results with the intersection of two balls.

Problem

• List Reconstruction
• Hamming Space
• Intersection of Balls
• Asymptotic Rate
• Associative Memories

Innovation

• list reconstruction
• Hamming space
• asymptotic rate
• intersection of balls
• coding theory