We show that polar codes asymptotically achieve the whole capacity-equivocation region for the wiretap channel when the wiretapper's channel is degraded with respect to the main channel, and the weak secrecy notion is used. Our coding scheme also achieves the capacity of the physically degraded receiver-orthogonal relay channel. We show simulation results for moderate block length for the binary erasure wiretap channel, comparing polar codes and two edge type LDPC codes.© 2010 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.QC 2011011

We derive the density evolution equations for non-binary low-density parity-check (LDPC) ensembles when transmission takes place over the binary erasure channel. We introduce ensembles defined with respect to the general linear group over the binary field.For these ensembles the density evolution equations can be written compactly. The density evolution for the general linear group helps us in understanding the density evolution for codes defined with respect to finite fields. We compute thresholds for different alphabet sizes for various LDPC ensembles. Surprisingly, the threshold is not a monotonic function of the alphabet size. We state the stability condition for non-binary LDPC ensembles over any binary memoryless symmetric channel. We also give upper bounds on the MAP thresholds for various non-binary ensembles based on EXIT curves and the area theorem.

Abstract-We consider the symmetric discrete memoryless relay channel with orthogonal receiver components and show that polar codes are suitable for decode-and-forward and compressand-forward relaying. In the first case we prove that polar codes are capacity achieving for the physically degraded relay channel; for stochastically degraded relay channels our construction provides an achievable rate. In the second case we construct sequences of polar codes that achieve the compress-and-forward rate by nesting polar codes for source compression into polar codes for channel coding. In both cases our constructions inherit most of the properties of polar codes. In particular, the encoding and decoding algorithms and the bound on the block error probability O(2 −N β ) which holds for any 0 < β < I. INTRODUCTIONThe relay channel, introduced by van der Meulen in [1], is an information theoretical model for cooperative communication in which a source wants to convey reliably a message to a destination with the help of a third node known as the relay. To determine the capacity of the relay channel in general is still an open problem. In [2] Cover and El Gamal established an outer bound on the capacity which is known as the cut-set bound and two coding strategies based on two different philosophies of information processing at the relay: decode-and-forward (DF) and compress-and-forward (CF). In DF the relay recovers the message transmitted by the source and forwards some information about it to the destination that complements the observation obtained through the source-destination link. In contrast, in CF the relay describes its raw channel observation. Since the destination has some side information (i.e. its own observation) this approach is connected to the Slepian-Wolf problems. Neither of these strategies (nor the combination of both [2]) is capacity achieving in general. Moreover, neither DF nor CF outperform the other in all the scenarios; as a rule of thumb DF performs better when the source-relay channel is good while CF is better when the quality of this channel is low [5].In the last decade there have been many research efforts to implement DF relaying in practice. The work has mainly focused on adapting capacity-approaching/achieving codes from

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