17 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
A High-Throughput Energy-Efficient Implementation of Successive-Cancellation Decoder for Polar Codes Using Combinational Logic
This paper proposes a high-throughput energy-efficient Successive
Cancellation (SC) decoder architecture for polar codes based on combinational
logic. The proposed combinational architecture operates at relatively low clock
frequencies compared to sequential circuits, but takes advantage of the high
degree of parallelism inherent in such architectures to provide a favorable
tradeoff between throughput and energy efficiency at short to medium block
lengths. At longer block lengths, the paper proposes a hybrid-logic SC decoder
that combines the advantageous aspects of the combinational decoder with the
low-complexity nature of sequential-logic decoders. Performance characteristics
on ASIC and FPGA are presented with a detailed power consumption analysis for
combinational decoders. Finally, the paper presents an analysis of the
complexity and delay of combinational decoders, and of the throughput gains
obtained by hybrid-logic decoders with respect to purely synchronous
architectures.Comment: 12 pages, 10 figures, 8 table
Systematic Encoding and Shortening of PAC Codes
Polarization adjusted convolutional (PAC) codes are a class of codes that combine channel polarization with convolutional coding. PAC codes are of interest for their high performance. This paper presents a systematic encoding and shortening method for PAC codes. Systematic encoding is important for lowering the bit-error rate (BER) of PAC codes. Shortening is important for adjusting the block length of PAC codes. It is shown that systematic encoding and shortening of PAC codes can be carried out in a unified framework