Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2510.22896

On the Arikan Transformations of Binary-Input Discrete Memoryless Channels

Yadong Jiao·Xiaoyan Cheng·Yuansheng Tang·Ming Xu
View original source ↗Read source document ↗Export metadata ↓

Abstract

The polar codes introduced by Arikan in 2009 achieve the capacity of binary-input discrete memoryless channels (BIDMCs) with low complexity encoding and decoding. Identifying the unreliable synthetic channels, generated by Arikan transformation during the construction of these polar codes, is crucial. Currently, because of the large size of the output alphabets of synthetic channels, there is no efficient and practical approach to evaluate their reliability in general. To tackle this problem, by converting the generation of synthetic channels in polar code construction into algebraic operations, in this paper we develop a method to characterize the synthetic channels as random switching channels of binary symmetric channels when the underlying channels are symmetric. Moreover, a lower bound for the average number of elements that possess the same likelihood ratio within the output alphabet of any synthetic channel generated in polar codes is also derived.

Explore related subjects

cs.ITmath.IT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yadong Jiao, Xiaoyan Cheng, Yuansheng Tang, Ming Xu. 2025-10-27. On the Arikan Transformations of Binary-Input Discrete Memoryless Channels. https://arxiv.org/abs/2510.22896

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fundamental Scaling Laws of Covert Communication in the Presence of Block Fading

Covert communication is the undetected transmission of sensitive information over a communication channel. In wireless communication systems, channel impairments such as signal fading present challenges in the effective implementation and analysis of covert communication systems. This paper generalizes early work in the covert communication field by considering asymptotic results for the number of bits that can be covertly transmitted in $n$ channel uses on a block fading channel. Critical to the investigation is characterizing the performance of optimal detectors at the adversary. Matching achievable and converse results are presented.

cs.IT↗
arXiv2026

Sequence Reconstruction over the Deletion Channel

In this paper, we consider the Levenshtein's sequence reconstruction problem in the case where the transmitted codeword is chosen from $\{0,1\}^n$ and the channel can delete up to $t$ symbols from the transmitted codeword. We determine the minimum number of channel outputs (assuming that they are distinct) required to reconstruct a list of size $\ell-1$ of candidate sequences, one of which corresponds to the original transmitted sequence. More specifically, we determine the maximum possible size of the intersection of $\ell \geq 3$ deletion balls of radius $t$ centered at $x_1, x_2, \dots, x_{\ell}$, where $x_i \in \{0,1\}^n$ for all $i \in \{1,2,\dots,\ell\}$ and $x_i \neq x_j$ for $i \neq j$, with $ n \geq t+\ell-1$ and $t \geq 1$.

cs.IT↗
arXiv2026

A generalization of the map $χ$

The mapping $ χ_n:\mathbb{F}_2^n \to \mathbb{F}_2^n$ defined by $y=χ_n(x)$ with $y_i = x_i + x_{i+1}x_{i+2} + x_{i+2}$, where the indices are computed modulo $n$, has been widely studied for its application in lightweight cryptography. In this paper, we generalize this mapping and completely characterize all these shift-invariant permutations of the form $y_i=x_{i+u}+x_{i+v}(x_{i+w}+a_i)$, where $0\le u, v, w<n$ and $a_i\in \mathbb{F}_2$, $1\le i\le n$.

cs.IT↗