From Polar to Reed-Muller Codes: a Technique to Improve the Finite-Length Performance

We explore the relationship between polar and RM codes and we describe a coding scheme which improves upon the performance of the standard polar code at practical block lengths. Our starting point is the experimental observation that RM codes have a smaller error probability than polar codes under MAP decoding. This motivates us to introduce a family of codes that "interpolates" between RM and polar codes, call this family ${\mathcal C}_{\rm inter} = \{C_{\alpha} : \alpha \in [0, 1]\}$, where $C_{\alpha} \big |_{\alpha = 1}$ is the original polar code, and $C_{\alpha} \big |_{\alpha = 0}$ is an RM code. Based on numerical observations, we remark that the error probability under MAP decoding is an increasing function of $\alpha$. MAP decoding has in general exponential complexity, but empirically the performance of polar codes at finite block lengths is boosted by moving along the family ${\mathcal C}_{\rm inter}$ even under low-complexity decoding schemes such as, for instance, belief propagation or successive cancellation list decoder. We demonstrate the performance gain via numerical simulations for transmission over the erasure channel as well as the Gaussian channel.Comment: 8 pages, 7 figures, in IEEE Transactions on Communications, 2014 and in ISIT'1

How to Achieve the Capacity of Asymmetric Channels

We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms. The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper. The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third paradigm is based on an idea of B\"ocherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published in IEEE Trans. Inform. Theor

Construction of Polar Codes with Sublinear Complexity

Consider the problem of constructing a polar code of block length $N$ for the transmission over a given channel $W$. Typically this requires to compute the reliability of all the $N$ synthetic channels and then to include those that are sufficiently reliable. However, we know from [1], [2] that there is a partial order among the synthetic channels. Hence, it is natural to ask whether we can exploit it to reduce the computational burden of the construction problem. We show that, if we take advantage of the partial order [1], [2], we can construct a polar code by computing the reliability of roughly a fraction $1/\log^{3/2} N$ of the synthetic channels. In particular, we prove that $N/\log^{3/2} N$ is a lower bound on the number of synthetic channels to be considered and such a bound is tight up to a multiplicative factor $\log\log N$. This set of roughly $N/\log^{3/2} N$ synthetic channels is universal, in the sense that it allows one to construct polar codes for any $W$, and it can be identified by solving a maximum matching problem on a bipartite graph. Our proof technique consists of reducing the construction problem to the problem of computing the maximum cardinality of an antichain for a suitable partially ordered set. As such, this method is general and it can be used to further improve the complexity of the construction problem in case a new partial order on the synthetic channels of polar codes is discovered.Comment: 9 pages, 3 figures, presented at ISIT'17 and submitted to IEEE Trans. Inform. Theor

Achieving Marton's Region for Broadcast Channels Using Polar Codes

This paper presents polar coding schemes for the 2-user discrete memoryless broadcast channel (DM-BC) which achieve Marton's region with both common and private messages. This is the best achievable rate region known to date, and it is tight for all classes of 2-user DM-BCs whose capacity regions are known. To accomplish this task, we first construct polar codes for both the superposition as well as the binning strategy. By combining these two schemes, we obtain Marton's region with private messages only. Finally, we show how to handle the case of common information. The proposed coding schemes possess the usual advantages of polar codes, i.e., they have low encoding and decoding complexity and a super-polynomial decay rate of the error probability. We follow the lead of Goela, Abbe, and Gastpar, who recently introduced polar codes emulating the superposition and binning schemes. In order to align the polar indices, for both schemes, their solution involves some degradedness constraints that are assumed to hold between the auxiliary random variables and the channel outputs. To remove these constraints, we consider the transmission of $k$ blocks and employ a chaining construction that guarantees the proper alignment of the polarized indices. The techniques described in this work are quite general, and they can be adopted to many other multi-terminal scenarios whenever there polar indices need to be aligned.Comment: 26 pages, 11 figures, accepted to IEEE Trans. Inform. Theory and presented in part at ISIT'1