Channel polarization with higher-order memory

Channel coding is one of the most fundamental problems regarding the nature of communication. Polar codes, developed by Arıkan, were the first demonstration of practical channel codes that provably achieve theoretical limits in a wide range of communication scenarios. Based on simple channel transformations which are called channel combining and splitting, Arıkan developed the idea of channel polarization which resulted in polar codes. In this dissertation we focus on the unique properties of polar codes and channel polarization. First, we investigate the channel-specific construction of polar codes, a point which discriminates polar codes from Reed-Muller (RM) codes. We obtain results showing the inherent effect of the underlying channel on the construction of polar codes and provide a bridge between polar and RM codes. Our results easily extend to obtaining a characterization for the rate of polarization as well. Next, we consider practical uses of polar codes for fading channels. We design a bit-interleaved polar-coded modulation scheme (BIPCM) by deriving a low complexity code-construction method and designing a lower complexity successive cancellation list decoder (SCLD). We compare the resultant BIPCM system with the existing solutions and show that it provides significant performance advantages. Finally, we generalize the channel polarization idea by changing channel combining and splitting operations. By introducing a memory order in the channel combining process we obtain a class of codes, including the original ones, that are parametrized by the memory order. We show that the new family of polar codes achieve the theoretical limits as well and they can also be used with lower complexity by increasing the memory order. We thereby complement Arıkan's conjecture that channel polarization is in fact a general phenomenon.