Construction of Polar Codes with Sublinear Complexity

TL;DR

This paper introduces a method to construct polar codes with significantly reduced computational complexity by leveraging a partial order among synthetic channels, requiring only a fraction of the channels to be evaluated.

Contribution

It demonstrates that polar code construction can be achieved by analyzing approximately N / log^{3/2} N synthetic channels, reducing complexity while maintaining universality.

Findings

Construction complexity is reduced to evaluating about N / log^{3/2} N channels.

The lower bound on channels to consider is tight up to a log-log factor.

The method is adaptable to new partial orders on synthetic channels.

Abstract

Consider the problem of constructing a polar code of block length $N$ for the transmission over a given channel $W$. Typically this requires to compute the reliability of all the $N$ synthetic channels and then to include those that are sufficiently reliable. However, we know from [1], [2] that there is a partial order among the synthetic channels. Hence, it is natural to ask whether we can exploit it to reduce the computational burden of the construction problem. We show that, if we take advantage of the partial order [1], [2], 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 new partial order on the synthetic channels of polar codes is discovered.