Bridging Hamming Distance Spectrum With Coset Cardinality Spectrum for Overlapped Arithmetic Codes

Distributed Source Coding (DSC), a scheme that encodes multiple correlated sources separately while decoding their bitstreams jointly, is an important branch of network information theory. Due to the advantages of shifting complexity burden from the encoder to the decoder and canceling the flow of data across terminals, DSC has potential applications in many scenarios, e.g., wireless sensor network, distributed genome data compression, etc. There are two forms (lossless and lossy) of DSC. Overlapped arithmetic codes, featured by overlapped intervals, are a variant of arithmetic codes that can implement distributed lossless compression, or the so-called Slepian-Wolf coding. For uniform binary sources, an overlapped arithmetic code is essentially a nonlinear many-to-one mapping that partitions source space into unequal-sized cosets. To analyze overlapped arithmetic codes, two theoretical tools have been proposed, i.e., 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 tightly bridging HDS with CCS. Specifically, HDS can be quickly and accurately calculated with CCS in some cases. In addition, the paper also proves the necessary and sufficient condition for the convergence of HDS and reveals the close relation between divergent HDS and polynomial division. All theoretical analyses are verified by experimental results.