NDLI: A packing lemma for polar codes

Content Provider	IEEE Xplore Digital Library
Author	Arikan, E.
Copyright Year	2015
Description	Author affiliation: Bilkent Univ., Ankara, Turkey (Arikan, E.)
Abstract	A packing lemma is proved using a setting where the channel is a binary-input discrete memoryless channel (X,w(y\|x),Y ), the code is selected at random subject to parity-check constraints, and the decoder is a joint typicality decoder. The ensemble is characterized by (i) a pair of fixed parameters (H,q) where H is a parity-check matrix and q is a channel input distribution and (ii) a random parameter S representing the desired parity values. For a code of length n, the constraint is sampled from $p_{S}(s)=Σx^{n}∈X^{n}$ $φ(s,x^{n})q^{n}(x^{n})$ where $φ(s,x^{n})$ is the indicator function of event {s = $x^{n}H^{T}\}$ and $q^{n}(x^{n})$ = $Π_{i=1}^{n}$ $q(x_{i}).$ Given S = s, the codewords are chosen conditionally independently from $p_{Xn\|S}(x^{n}\|s)$ ∝ $φ(s,x^{n})q^{n}(x^{n}).$ It is shown that the probability of error for this ensemble decreases exponentially in n provided the rate R is kept bounded away from I(X;Y )-1 over $nI(S;Y^{n})$ with (X,Y ) ~ q(x)w(y\|x) and $(S,Y^{n})$ ~ $p_{S}(s)Σx^{n}$ $p_{Xn\|S}(x^{n}\|s)Π_{i=1}^{n}=w(y_{i}\|x_{i}).$ In the special case where H is the parity-check matrix of a standard polar code, it is shown that the rate penalty 1 over $nI(S;Y^{n})$ vanishes as n increases. The paper also discusses the relation between ordinary polar codes and random codes based on polar parity-check matrices.
Starting Page	2441
Ending Page	2445
File Size	159002
Page Count	5
File Format	PDF
ISSN	21578117
e-ISBN	9781467377041
DOI	10.1109/ISIT.2015.7282894
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2015-06-14
Publisher Place	China
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Encoding Decoding Joints Standards Parity check codes Memoryless systems
Content Type	Text
Resource Type	Article

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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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