Symmetrical Multilevel Diversity Coding and Subset Entropy Inequalities

TL;DR

This paper revisits and extends the optimality proofs of superposition coding in symmetrical multilevel diversity coding (SMDC), introducing new subset entropy inequalities and applying them to more complex scenarios with access or secrecy constraints.

Contribution

It introduces a new sliding-window subset entropy inequality and uses it to strengthen the understanding of superposition coding's optimality in SMDC and related models.

Findings

New sliding-window subset entropy inequality established.

Superposition coding proven optimal for minimum sum rate with weaker requirements.

Extended optimality results to models with additional access or secrecy constraints.

Abstract

Symmetrical multilevel diversity coding (SMDC) is a classical model for coding over distributed storage. In this setting, a simple separate encoding strategy known as superposition coding was shown to be optimal in terms of achieving the minimum sum rate (Roche, Yeung, and Hau, 1997) and the entire admissible rate region (Yeung and Zhang, 1999) of the problem. The proofs utilized carefully constructed induction arguments, for which the classical subset entropy inequality of Han (1978) played a key role. This paper includes two parts. In the first part the existing optimality proofs for classical SMDC are revisited, with a focus on their connections to subset entropy inequalities. First, a new sliding-window subset entropy inequality is introduced and then used to establish the optimality of superposition coding for achieving the minimum sum rate under a weaker source-reconstruction requirement. Second, a subset entropy inequality recently proved by Madiman and Tetali (2010) is used to develop a new structural understanding to the proof of Yeung and Zhang on the optimality of superposition coding for achieving the entire admissible rate region. Building on the connections between classical SMDC and the subset entropy inequalities developed in the first part, in the second part the optimality of superposition coding is further extended to the cases where there is either an additional all-access encoder (SMDC-A) or an additional secrecy constraint (S-SMDC).