The Explicit Coding Rate Region of Symmetric Multilevel Diversity Coding

Abstract

It is well known that superposition coding, namely separately encoding the independent sources, is optimal for symmetric multilevel diversity coding (SMDC) (Yeung-Zhang 1999) for any L ≥ 2, where L 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 L = 5. In this paper, for any fixed L, 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 L. 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.

Weakly Secure Symmetric Multilevel Diversity Coding

“…This is currently under our investigation. APPENDIX A SOME LEMMAS/THEOREMS FROM [3], [4] In the following lemmas and theorem, we assume λ ∈ R L (…”

“…For any permutation ω on {1, 2, · · · , L}, denote λ ω(1) , λ ω(2) , · · · , λ ω(L) by ω(λ). Lemma 2 in [4]: f α ω(λ) = f α (λ) for any α ∈ L. Lemma 5 in [4]: Let λ 1 = (λ 1,1 , λ 1,2 , · · · , λ 1,L ) and λ 2 = (λ 2,1 , λ 2,2 , · · · , λ 2,L ) be two ordered vectors such that λ 1,1 > λ 2,1 and λ 1,i = λ 2,i for all…”

“…Similar to Lemma 6 in [4], the following lemma gives a sufficient condition of redundancy in the characterization of the rate region.…”

“…Furthermore, an α-resolution is called optimal if it achieves the optimal value f α (λ). In the following proof, we will take advantage of some lemmas and theorems from [3] and [4], which are enclosed in Appendix A for convenience.…”

“…Guo It was shown [1], [3] that separately encoding these independent messages, referred to as superposition coding, is optimal in terms of achieving the entire rate region. The characterization of the coding rate region therein involves implicit and uncountably many inequalities, and an explicit characterization of the coding rate region was recently obtained [4]. The problem has also been extended and generalized, e.g., to allow node regeneration [5] and to allow asymmetric decoders [6].…”