Symmetrical Multilevel Diversity Coding and Subset Entropy Inequalities

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 and the entire admissible rate region of the problem. The proofs utilized carefully constructed induction arguments, for which the classical subset entropy inequality played a key role. This paper consists of 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. Initially, 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. Finally, a subset entropy inequality recently proved by Madiman and Tetali is used to develop a new structural understanding of the work 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 extended to the cases where there is either an additional all-access encoder or an additional secrecy constraint.