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.

翻译：随着基于DNA的数据存储系统的最新进展，对多个$q$元汉明球交的大小的研究兴趣日益增长。我们提出了一个精确公式，用于计算$q$元字母表上$s个半径各异的汉明球交的基数。已知在一般情况下，仅凭汉明球中心点之间的距离不足以确定交的大小。基于我们的公式，我们能够找到中心点更精细的结构特性，从而确定交的确切大小。此外，我们还分析了当$n$足够大时交的大小。当$s=3$时，我们给出了（对于所有$q\ge 2$、$q\neq 6$且$n$足够大的情况）在汉明球中心点具有给定最小距离时获得最大交大小的充要条件，并展示了如何利用我们的通用公式进行计算。