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

TL;DR

This paper introduces the symmetric convex ordering, a new partial order for B-DMCs, demonstrating its preservation under polar transforms and its implications for ordering polar code information sets, especially in asymmetric channels.

Contribution

It proposes the symmetric convex ordering for B-DMCs, showing its relation to existing orders and its utility in analyzing polar codes over various channels.

Findings

Symmetric convex ordering is preserved under polar transforms.

For symmetric channels, it aligns with stochastic degradation ordering.

A weaker order applies to asymmetric channels.

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.