A partial order for the synthesized channels of a polar code

A partial order for the synthesized channels W
_N
⁽ⁱ⁾
of a polar code is presented that is independent of the underlying binary-input channel W. The partial order is based on the observation that W
_N
^(j)
is stochastically degraded to W
_N
⁽ⁱ⁾
if j is obtained by swapping a more significant 1 with a less significant 0 in the binary expansion of i. We derive an efficient representation of the partial order, the so-called covering relation. The partial order is then combined with another partial order from the literature that is also independent of W. Finally, we give some remarks on how this combined partial order can be used to simplify code construction of polar codes.