Polar Coding in Certain New Transmission Environments

Polar codes, introduced by Arikan in 2009, have attracted considerable interest as an asymptotically capacity-achieving code with sufficient performance advantages to merit inclusion in the 5G standard. Polar codes are constructed directly from an explicit model of the communication channel, so their performance is dependent on a detailed understanding of the transmission environment. We partially remove a basic assumption in coding theory that channels are identical and independent by extending polar codes to several types of channels with memory, including periodic Markov processes and Information Regular processes. In addition, we consider modifications to the polar code construction so that the inclusion of a shared secret in the frozen set naturally produces encryption via one-time pad. We describe one such modification in terms of the achievable frozen sets which are compatible with the polar code automorphism group. We then provide a partial characterization of these frozen sets using an explicit construction for the Linear Extension Diameter of channel entropies.