The Symmetric Convex Ordering: A Novel Partial Order for B-DMCs Ordering the Information Sets of Polar Codes

Abstract

In this paper, we propose a novel partial order for binary discrete memoryless channels that we call the symmetric convex ordering. We show that Ar{\i}kan's polar transform preserves 'symmetric convex orders'. Furthermore, we show that 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, a strictly weaker partial order is obtained when at least one of the channels is asymmetric. In between, we also discuss two tools which can be useful for verifying this ordering: a criterion known as the cut criterion and channel symmetrization. Finally, we discuss potential applications of the results to polar coding over non-stationary channels.