Overlapped arithmetic codes, featured by overlapped intervals, are a variant of arithmetic codes that can be used to implement Slepian-Wolf coding. To analyze overlapped arithmetic codes, we have proposed two theoretical tools: Coset Cardinality Spectrum (CCS) and Hamming Distance Spectrum (HDS). The former describes how source space is partitioned into cosets (equally or unequally), and the latter describes how codewords are structured within each coset (densely or sparsely). However, until now, these two tools are almost parallel to each other, and it seems that there is no intersection between them. The main contribution of this paper is bridging HDS with CCS through a rigorous mathematical proof. Specifically, HDS can be quickly and accurately calculated with CCS in some cases. All theoretical analyses are perfectly verified by simulation results.

翻译：重叠算术码是一种通过重叠区间实现的算术码变体，可用于实现Slepian-Wolf编码。为分析重叠算术码，我们提出了两种理论工具：陪集基数谱(CCS)和汉明距离谱(HDS)。前者描述信源空间如何被划分为陪集（等分或不等分），后者描述每个陪集内码字的分布结构（密集或稀疏）。然而，迄今为止这两种工具几乎相互独立，似乎不存在任何交集。本文的主要贡献在于通过严谨的数学证明，将HDS与CCS建立联系。具体而言，在特定情况下可通过CCS快速精确地计算HDS。所有理论分析均通过仿真结果得到完美验证。