2,198 research outputs found
Varentropy Decreases Under the Polar Transform
We consider the evolution of variance of entropy (varentropy) in the course
of a polar transform operation on binary data elements (BDEs). A BDE is a pair
consisting of a binary random variable and an arbitrary side
information random variable . The varentropy of is defined as the
variance of the random variable . A polar transform of
order two is a certain mapping that takes two independent BDEs and produces two
new BDEs that are correlated with each other. It is shown that the sum of the
varentropies at the output of the polar transform is less than or equal to the
sum of the varentropies at the input, with equality if and only if at least one
of the inputs has zero varentropy. This result is extended to polar transforms
of higher orders and it is shown that the varentropy decreases to zero
asymptotically when the BDEs at the input are independent and identially
distributed.Comment: Presented in part at ISIT 2014. Accepted for publication in the IEEE
Trans. Inform. Theory, March 201
A Packing Lemma for Polar Codes
A packing lemma is proved using a setting where the channel is a binary-input
discrete memoryless channel , 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 where is a parity-check matrix and is a channel
input distribution and (ii) a random parameter representing the desired
parity values. For a code of length , the constraint is sampled from where is the
indicator function of event and . Given , the codewords are chosen conditionally
independently from . It is shown
that the probability of error for this ensemble decreases exponentially in
provided the rate is kept bounded away from
with and . In the special case where is the parity-check
matrix of a standard polar code, it is shown that the rate penalty
vanishes as increases. The paper also discusses the
relation between ordinary polar codes and random codes based on polar
parity-check matrices.Comment: 5 pages. To be presented at 2015 IEEE International Symposium on
Information Theory, June 14-19, 2015, Hong Kong. Minor corrections to v
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