The Explicit Coding Rate Region of Symmetric Multilevel Diversity Coding

It is well known that <italic>superposition coding</italic>, namely separately encoding the independent sources, is optimal for symmetric multilevel diversity coding (SMDC) (Yeung-Zhang 1999) for any <inline-formula> <tex-math notation="LaTeX">$L\geq 2$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> is the number of levels of the coding system. However, the characterization of the coding rate region therein involves uncountably many linear inequalities and the constant term (i.e., the lower bound) in each inequality is given in terms of the solution of a linear optimization problem. Thus this implicit characterization of the coding rate region does not enable the determination of the achievability of a given rate tuple. In principle, the achievability of a given rate tuple can be determined by direct computation, but the complexity is prohibitive even for <inline-formula> <tex-math notation="LaTeX">$L=5$ </tex-math></inline-formula>. In this paper, for any fixed <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>, we obtain in closed form a finite set of linear inequalities for characterizing the coding rate region. We further show by the symmetry of the problem that only a much smaller subset of this finite set of inequalities needs to be verified in determining the achievability of a given rate tuple. Yet, the cardinality of this smaller set grows at least exponentially fast with <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>. We also present a subset entropy inequality, which together with our explicit characterization of the coding rate region, is sufficient for proving the optimality of superposition coding.