Strong Converse Theorems for Classes of Multimessage Multicast Networks: A R\'enyi Divergence Approach

This paper establishes that the strong converse holds for some classes of discrete memoryless multimessage multicast networks (DM-MMNs) whose corresponding cut-set bounds are tight, i.e., coincide with the set of achievable rate tuples. The strong converse for these classes of DM-MMNs implies that all sequences of codes with rate tuples belonging to the exterior of the cut-set bound have average error probabilities that necessarily tend to one (and are not simply bounded away from zero). Examples in the classes of DM-MMNs include wireless erasure networks, DM-MMNs consisting of independent discrete memoryless channels (DMCs) as well as single-destination DM-MMNs consisting of independent DMCs with destination feedback. Our elementary proof technique leverages properties of the R\'enyi divergence.Comment: Submitted to IEEE Transactions on Information Theory, Jul 18, 2014. Revised on Jul 31, 201

Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel

This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The scaling exponent $\mu$ of polar codes for a memoryless channel $q_{Y|X}$ with capacity $I(q_{Y|X})$ characterizes the closest gap between the capacity and non-asymptotic achievable rates in the following way: For a fixed $\varepsilon \in (0, 1)$, the gap between the capacity $I(q_{Y|X})$ and the maximum non-asymptotic rate $R_n^*$ achieved by a length-$n$ polar code with average error probability $\varepsilon$ scales as $n^{-1/\mu}$, i.e., $I(q_{Y|X})-R_n^* = \Theta(n^{-1/\mu})$. It is well known that the scaling exponent $\mu$ for any binary-input memoryless channel (BMC) with $I(q_{Y|X})\in(0,1)$ is bounded above by $4.714$, which was shown by an explicit construction of polar codes. Our main result shows that $4.714$ remains to be a valid upper bound on the scaling exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of $O(n^{-1/\mu}\sqrt{\log n})$ by using an input alphabet consisting of $n$ constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of $n$ constellations can be achieved within a gap of $O(n^{-1/\mu}\log n)$ by using a superposition of $\log n$ binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as $n$ grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel.Comment: 24 page