Distributed Channel Synthesis

Two familiar notions of correlation are rediscovered as the extreme operating points for distributed synthesis of a discrete memoryless channel, in which a stochastic channel output is generated based on a compressed description of the channel input. Wyner's common information is the minimum description rate needed. However, when common randomness independent of the input is available, the necessary description rate reduces to Shannon's mutual information. This work characterizes the optimal trade-off between the amount of common randomness used and the required rate of description. We also include a number of related derivations, including the effect of limited local randomness, rate requirements for secrecy, applications to game theory, and new insights into common information duality. Our proof makes use of a soft covering lemma, known in the literature for its role in quantifying the resolvability of a channel. The direct proof (achievability) constructs a feasible joint distribution over all parts of the system using a soft covering, from which the behavior of the encoder and decoder is inferred, with no explicit reference to joint typicality or binning. Of auxiliary interest, this work also generalizes and strengthens this soft covering tool.Comment: To appear in IEEE Trans. on Information Theory (submitted Aug., 2012, accepted July, 2013), 26 pages, using IEEEtran.cl

Polar codes with a stepped boundary

We consider explicit polar constructions of blocklength $n\rightarrow\infty$ for the two extreme cases of code rates $R\rightarrow1$ and $R\rightarrow0.$ For code rates $R\rightarrow1,$ we design codes with complexity order of $n\log n$ in code construction, encoding, and decoding. These codes achieve the vanishing output bit error rates on the binary symmetric channels with any transition error probability $p\rightarrow 0$ and perform this task with a substantially smaller redundancy $(1-R)n$ than do other known high-rate codes, such as BCH codes or Reed-Muller (RM). We then extend our design to the low-rate codes that achieve the vanishing output error rates with the same complexity order of $n\log n$ and an asymptotically optimal code rate $R\rightarrow0$ for the case of $p\rightarrow1/2.$Comment: This article has been submitted to ISIT 201