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

TL;DR

This paper establishes a rigorous mathematical connection between Hamming Distance Spectrum and Coset Cardinality Spectrum in overlapped arithmetic codes, enabling efficient analysis and verification through simulations.

Contribution

It introduces a novel theoretical link between HDS and CCS, allowing HDS to be computed from CCS in certain scenarios, advancing the analysis of overlapped arithmetic codes.

Findings

HDS can be accurately derived from CCS in specific cases

Theoretical analysis is validated by simulation results

Bridging HDS and CCS enhances understanding of overlapped arithmetic codes

Abstract

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.