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In this paper, we propose a novel partial order for binary discretememoryless channels that we call the symmetric convex ordering. We show thatAr{\i}kan's polar transform preserves 'symmetric convex orders'. Furthermore,we show that while for symmetric channels this ordering turns out to beequivalent to the stochastic degradation ordering already known to order theinformation sets of polar codes, a strictly weaker partial order is obtainedwhen at least one of the channels is asymmetric. In between, we also discusstwo tools which can be useful for verifying this ordering: a criterion known asthe cut criterion and channel symmetrization. Finally, we discuss potentialapplications of the results to polar coding over non-stationary channels.

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