A novel partial order for the information sets of polar codes over B-DMCs

We study partial orders on the information sets of polar codes designed for binary discrete memoryless channels. We show that the polar transform defined by Arikan preserves `symmetric convex/concave orders´. While for symmetric channels this ordering turns out to be equivalent to the stochastic degradation ordering already known to order the information sets of polar codes, we show that a strictly weaker partial order is obtained when at least one of the channels is asymmetric. We also discuss two tools which can be useful for verifying this ordering: a criterion known as the cut criterion and channel symmetrization.