---
_id: '6663'
abstract:
- lang: eng
text: 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.
author:
- first_name: Marco
full_name: Mondelli, Marco
id: 27EB676C-8706-11E9-9510-7717E6697425
last_name: Mondelli
orcid: 0000-0002-3242-7020
- first_name: Hamed
full_name: Hassani, Hamed
last_name: Hassani
- first_name: Rudiger
full_name: Urbanke, Rudiger
last_name: Urbanke
citation:
ama: Mondelli M, Hassani H, Urbanke R. Construction of polar codes with sublinear
complexity. *IEEE*. 2019;65(5):2782-2791. doi:10.1109/tit.2018.2889667
apa: Mondelli, M., Hassani, H., & Urbanke, R. (2019). Construction of polar
codes with sublinear complexity. *IEEE*. IEEE. https://doi.org/10.1109/tit.2018.2889667
chicago: Mondelli, Marco, Hamed Hassani, and Rudiger Urbanke. “Construction of Polar
Codes with Sublinear Complexity.” *IEEE*. IEEE, 2019. https://doi.org/10.1109/tit.2018.2889667.
ieee: M. Mondelli, H. Hassani, and R. Urbanke, “Construction of polar codes with
sublinear complexity,” *IEEE*, vol. 65, no. 5. IEEE, pp. 2782–2791, 2019.
ista: Mondelli M, Hassani H, Urbanke R. 2019. Construction of polar codes with sublinear
complexity. IEEE. 65(5), 2782–2791.
mla: Mondelli, Marco, et al. “Construction of Polar Codes with Sublinear Complexity.”
*IEEE*, vol. 65, no. 5, IEEE, 2019, pp. 2782–91, doi:10.1109/tit.2018.2889667.
short: M. Mondelli, H. Hassani, R. Urbanke, IEEE 65 (2019) 2782–2791.
date_created: 2019-07-23T07:32:57Z
date_published: 2019-05-01T00:00:00Z
date_updated: 2023-02-23T12:50:20Z
day: '01'
doi: 10.1109/tit.2018.2889667
extern: '1'
external_id:
arxiv:
- '1612.05295'
intvolume: ' 65'
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1612.05295
month: '05'
oa: 1
oa_version: Preprint
page: 2782-2791
publication: IEEE
publication_status: published
publisher: IEEE
quality_controlled: '1'
related_material:
record:
- id: '6729'
relation: earlier_version
status: public
status: public
title: Construction of polar codes with sublinear complexity
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 65
year: '2019'
...