Concatenated Polar Codes

Polar codes have attracted much recent attention as the first codes with low computational complexity that provably achieve optimal rate-regions for a large class of information-theoretic problems. One significant drawback, however, is that for current constructions the probability of error decays sub-exponentially in the block-length (more detailed designs improve the probability of error at the cost of significantly increased computational complexity \cite{KorUS09}). In this work we show how the the classical idea of code concatenation -- using "short" polar codes as inner codes and a "high-rate" Reed-Solomon code as the outer code -- results in substantially improved performance. In particular, code concatenation with a careful choice of parameters boosts the rate of decay of the probability of error to almost exponential in the block-length with essentially no loss in computational complexity. We demonstrate such performance improvements for three sets of information-theoretic problems -- a classical point-to-point channel coding problem, a class of multiple-input multiple output channel coding problems, and some network source coding problems

Space-Time Signal Design for Multilevel Polar Coding in Slow Fading Broadcast Channels

Slow fading broadcast channels can model a wide range of applications in wireless networks. Due to delay requirements and the unavailability of the channel state information at the transmitter (CSIT), these channels for many applications are non-ergodic. The appropriate measure for designing signals in non-ergodic channels is the outage probability. In this paper, we provide a method to optimize STBCs based on the outage probability at moderate SNRs. Multilevel polar coded-modulation is a new class of coded-modulation techniques that benefits from low complexity decoders and simple rate matching. In this paper, we derive the outage optimality condition for multistage decoding and propose a rule for determining component code rates. We also derive an upper bound on the outage probability of STBCs for designing the set-partitioning-based labelling. Finally, due to the optimality of the outage-minimized STBCs for long codes, we introduce a novel method for the joint optimization of short-to-moderate length polar codes and STBCs

Multilevel Polarization of Polar Codes Over Arbitrary Discrete Memoryless Channels

It is shown that polar codes achieve the symmetric capacity of discrete memoryless channels with arbitrary input alphabet sizes. It is shown that in general, channel polarization happens in several, rather than only two levels so that the synthesized channels are either useless, perfect or "partially perfect". Any subset of the channel input alphabet which is closed under addition, induces a coset partition of the alphabet through its shifts. For any such partition of the input alphabet, there exists a corresponding partially perfect channel whose outputs uniquely determine the coset to which the channel input belongs. By a slight modification of the encoding and decoding rules, it is shown that perfect transmission of certain information symbols over partially perfect channels is possible. Our result is general regarding both the cardinality and the algebraic structure of the channel input alphabet; i.e we show that for any channel input alphabet size and any Abelian group structure on the alphabet, polar codes are optimal. It is also shown through an example that polar codes when considered as group/coset codes, do not achieve the capacity achievable using coset codes over arbitrary channels

Information Design for Strategic Coordination of Autonomous Devices with Non-Aligned Utilities

In this paper, we investigate the coordination of autonomous devices with non-aligned utility functions. Both encoder and decoder are considered as players, that choose the encoding and the decoding in order to maximize their long-run utility functions. The topology of the point-to-point network under investigation, suggests that the decoder implements a strategy, knowing in advance the strategy of the encoder. We characterize the encoding and decoding functions that form an equilibrium, by using empirical coordination. The equilibrium solution is related to an auxiliary game in which both players choose some conditional distributions in order to maximize their expected utilities. This problem is closely related to the literature on "Information Design" in Game Theory. We also characterize the set of posterior distributions that are compatible with a rate-limited channel between the encoder and the decoder. Finally, we provide an example of non-aligned utility functions corresponding to parallel fading multiple access channels.Comment: IEEE Proc. of the Fifty-fourth Annual Allerton Conference Allerton House, UIUC, Illinois, USA September 27 - 30, 201