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.

I. Introduction

Symmetrical multilevel diversity coding (SMDC) is a classical model for coding over distributed storage. The problem was first introduced by Roche [1] and Yeung [2]. In this setting, there are a total of independent discrete memoryless sources , where the importance of the source is assumed to decrease with the subscript . The sources are to be encoded by a total of randomly accessible encoders. The goal of encoding is to ensure that the number of sources that can be nearly perfectly reconstructed grows with the number of available encoder outputs at the decoder. More specifically, denote by the set of accessible encoders. The realization of is unknown a priori at the encoders. However, the sources need to be nearly perfectly reconstructed whenever . The word “symmetrical” here refers to the fact that the sources that need to be nearly perfectly reconstructed depend on the set of accessible encoders only via its cardinality. The rate allocations at different encoders, however, can be different and are not necessarily symmetrical.