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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… (More)

Consider transmission of a polar code of block length N and rate R over a binary memoryless symmetric channel W and let P e be the error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship among the parameters R, N , P e , and the quality of the channel W quantified by its capacity I(W)… (More)

—We show that Reed-Muller codes achieve capacity under maximum a posteriori bit decoding for transmission over the binary erasure channel for all rates 0 < R < 1. The proof is generic and applies to other codes with sufficient amount of symmetry as well. The main idea is to combine the following observations: (i) monotone functions experience a sharp… (More)

We discuss coding techniques that allow reliable transmission up to the capacity of a discrete memoryless asymmetric channel. Some of the techniques are well-known and go back sixty years ago, some are recent, and others are new. We take the point of view of modern coding theory and we discuss how recent advances in coding for symmetric channels help in… (More)

—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… (More)

—Motivated by the significant performance gains which polar codes experience under successive cancellation list decoding, their scaling exponent is studied as a function of the list size. In particular, the error probability is fixed and the trade-off between block length and back-off from capacity is analyzed. A lower bound is provided on the error… (More)

—The question whether RM codes are capacity-achieving is a long-standing open problem in coding theory that was recently answered in the affirmative for transmission over erasure channels [1], [2]. Remarkably, the proof does not rely on specific properties of RM codes, apart from their symmetry. Indeed, the main technical result consists in showing that any… (More)

—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… (More)