Construction of Polar Codes With Sublinear Complexity

Consider the problem of constructing a polar code of block length
$N$
for a given transmission channel
$W$
. Previous approaches require one to compute the reliability of the
$N$
synthetic channels and then use only those that are sufficiently reliable. However, we know from two independent works by Schürch and by Bardet
et al.
that the synthetic channels are partially ordered with respect to degradation. Hence, it is natural to ask whether the partial order can be exploited to reduce the computational burden of the construction problem. We show that, if we take advantage of the partial order, we can construct a polar code by computing the reliability of roughly a fraction
$1/\log ^{3/2} N$
of the synthetic channels. In particular, we prove that
$N/\log ^{3/2} N$
is a lower bound on the number of synthetic channels to be considered and such a bound is tight up to a multiplicative factor
$\log \log N$
. This set of roughly
$N/\log ^{3/2} N$
synthetic channels is universal, in the sense that it allows one to construct polar codes for any
$W$
, and it can be identified by solving a maximum matching problem on a bipartite graph. Our proof technique consists of reducing the construction problem to the problem of computing the maximum cardinality of an antichain for a suitable partially ordered set. As such, this method is general, and it can be used to further improve the complexity of the construction problem, in case a refined partial order on the synthetic channels of polar codes is discovered.