Channel Polarization and Polar Codes

Polar coding is a new technique introduced by Arıkan [1] based on a phenomenon called channel polarization. Polar codes are appealing as an error correction method since they are proved to achieve the symmetric capacity of any B-DMC using low complexity encoders and decoders, and their block error probability is shown to decrease exponentially in the square root of the block length. In fact, two basic channel transformations lie at the heart of channel polarization. The recursive applications of these transformations result in channel polarization which refers to the fact that the channels synthesized in these transformations become in the limit either almost perfect or completely noisy. This is shown by analyzing channel parameters such as the symmetric capacity and the Bhattacharya parameter. An important characteristic of polar codes is that they are channel specific codes. For that particular reason, the channel over which we communicate information should be considered during the code design. However, the binary erasure channel and the binary symmetric channel stand out as extremal channels among all B-DMCs in terms of the evolution of the channel symmetric capacity and the Bhattacharyya parameter under the basic channel transformations. In this work, we generalize this extremality result to a more general parameter E0(ρ,W ), defined by Gallager [2] as part of the random coding exponent, for a B-DMC W with uniform input distribution. We show that the binary erasure channel and the binary symmetric channel are also extremal with respect to the evolution of the parameter E0(ρ,W ) under the basic channel transformations. Then, we conjecture an inequality between E0(ρ,W ), E0(ρ,W), and E0(ρ,W ). In the process, we note that the function E0(ρ,W )/ρ is interpreted as a general measure of information using Rényi’s entropy functions. Moreover, we discuss an application on the compound capacity of polar codes under successive cancellation decoding where the extremality of the binary erasure channel is used to derive a lower bound. We also provide a discussion on the compound capacity of linear codes which includes polar codes as sub-class. We show that while linear codes achieving the compound capacity of symmetric channels exist, the existing results on the compound capacity of polar codes decoded using a successive cancellation decoder shows that, in general, the compound capacity of symmetric channels is not achieved by polar codes. In addition, independently of channel polarization, we undertake a study of another channel property: the random coding exponent in parametric form. Note that this parametric description is a function of E0(ρ,W ) and the rate R(ρ,W ) = ∂E0(ρ,W )/∂ρ. We extend the binary erasure channel and the binary symmetric channel extremality result in [3] in terms of E0(ρ,W ) and the rate R(ρ,W ) to the case where we have different ρ values, i.e., E0(ρ1,W ) and R(ρ2,W ).