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.

翻译：在漢明空間中計算多個球交集的基數問題，因其在列表重建問題與關聯記憶體資訊檢索中的應用，近期備受關注。過往研究多聚焦於當碼字長度 n 趨於無窮大時，每個球的半徑 r 與球心之間距離 k 為固定值的情況。本研究則著重探討 r = αn 且 k = βn（其中 α 與 β 為常數）的情況，並計算三個球交集基數的最大漸近速率。我們以兩個參數 α 與 β 的函數形式，給出此最大漸近速率，同時提供數值結果，並將這些結果與兩個球交集的情況進行比較。