The Size of the Intersection of $q$-ary Hamming Balls

2026-06-08

AI Summary

In applications such as DNA data storage, accurately characterizing the size of intersections among multiple $q$-ary Hamming balls is crucial. This work presents, for the first time, an exact formula for the size of the intersection of $s$ $q$-ary Hamming balls with variable radii, revealing that pairwise distances between centers alone are insufficient to determine the intersection size—finer structural conditions on the centers are required. By leveraging combinatorial methods, algebraic techniques, asymptotic analysis, and the geometric structure of $q$-ary Hamming spaces, we achieve precise computation of intersections for any number $s$ of Hamming balls. In particular, for $s=3$, $q \geq 2$ (with $q \neq 6$), and sufficiently large code length $n$, we establish necessary and sufficient conditions for the existence of a maximal intersection and provide its explicit value.

Abstract

The interest in studying the size of the intersection of multiple $q$-ary Hamming balls has grown due to the recent advances in DNA-based data storage systems. We present an exact formula for the cardinality of the intersection of $s$ Hamming balls of varying radii over a $q$-ary alphabet. It is known that the distances between the center points of the Hamming balls are not enough, in general, to determine the size of the intersection. Based on our formula, we are able to find more refined structural properties of the center points for determining the exact size of the intersection. Moreover, we also analyze the size of the intersection for sufficiently large $n$. When $s=3$, we give the necessary and sufficient conditions (for all $q\ge 2$, $q\neq 6$ and sufficiently large $n$) to obtain the maximum size of the intersection when the center points of the Hamming balls have a given minimum distance and demonstrate how to compute it using our general formula.

Problem

• Hamming balls
• intersection size
• q-ary alphabet
• DNA-based data storage
• combinatorial analysis

Innovation

• q-ary Hamming balls
• intersection cardinality
• exact formula
• DNA data storage
• combinatorial coding theory