13 research outputs found
Concentration inequalities for order statistics
This note describes non-asymptotic variance and tail bounds for order
statistics of samples of independent identically distributed random variables.
Those bounds are checked to be asymptotically tight when the sampling
distribution belongs to a maximum domain of attraction. If the sampling
distribution has non-decreasing hazard rate (this includes the Gaussian
distribution), we derive an exponential Efron-Stein inequality for order
statistics: an inequality connecting the logarithmic moment generating function
of centered order statistics with exponential moments of Efron-Stein
(jackknife) estimates of variance. We use this general connection to derive
variance and tail bounds for order statistics of Gaussian sample. Those bounds
are not within the scope of the Tsirelson-Ibragimov-Sudakov
Gaussian concentration inequality. Proofs are elementary and combine
R\'enyi's representation of order statistics and the so-called entropy approach
to concentration inequalities popularized by M. Ledoux.Comment: 13 page
Almost Optimal Scaling of Reed-Muller Codes on BEC and BSC Channels
Consider a binary linear code of length , minimum distance
, transmission over the binary erasure channel with parameter
or the binary symmetric channel with parameter , and block-MAP decoding. It was shown by Tillich and Zemor that in
this case the error probability of the block-MAP decoder transitions "quickly"
from to for any if the minimum distance is
large. In particular the width of the transition is of order
. We strengthen this result by showing that under
suitable conditions on the weight distribution of the code, the transition
width can be as small as , for any ,
even if the minimum distance of the code is not linear. This condition applies
e.g., to Reed-Mueller codes. Since is the smallest
transition possible for any code, we speak of "almost" optimal scaling. We
emphasize that the width of the transition says nothing about the location of
the transition. Therefore this result has no bearing on whether a code is
capacity-achieving or not. As a second contribution, we present a new estimate
on the derivative of the EXIT function, the proof of which is based on the
Blowing-Up Lemma.Comment: Submitted to ISIT 201
Algebraic Properties of Polar Codes From a New Polynomial Formalism
Polar codes form a very powerful family of codes with a low complexity
decoding algorithm that attain many information theoretic limits in error
correction and source coding. These codes are closely related to Reed-Muller
codes because both can be described with the same algebraic formalism, namely
they are generated by evaluations of monomials. However, finding the right set
of generating monomials for a polar code which optimises the decoding
performances is a hard task and channel dependent. The purpose of this paper is
to reveal some universal properties of these monomials. We will namely prove
that there is a way to define a nontrivial (partial) order on monomials so that
the monomials generating a polar code devised fo a binary-input symmetric
channel always form a decreasing set.
This property turns out to have rather deep consequences on the structure of
the polar code. Indeed, the permutation group of a decreasing monomial code
contains a large group called lower triangular affine group. Furthermore, the
codewords of minimum weight correspond exactly to the orbits of the minimum
weight codewords that are obtained from (evaluations) of monomials of the
generating set. In particular, it gives an efficient way of counting the number
of minimum weight codewords of a decreasing monomial code and henceforth of a
polar code.Comment: 14 pages * A reference to the work of Bernhard Geiger has been added
(arXiv:1506.05231) * Lemma 3 has been changed a little bit in order to prove
that Proposition 7.1 in arXiv:1506.05231 holds for any binary input symmetric
channe
Recursive projection-aggregation decoding of Reed-Muller codes
We propose a new class of efficient decoding algorithms for Reed-Muller (RM)
codes over binary-input memoryless channels. The algorithms are based on
projecting the code on its cosets, recursively decoding the projected codes
(which are lower-order RM codes), and aggregating the reconstructions (e.g.,
using majority votes). We further provide extensions of the algorithms using
list-decoding.
We run our algorithm for AWGN channels and Binary Symmetric Channels at the
short code length () regime for a wide range of code rates.
Simulation results show that in both low code rate and high code rate regimes,
the new algorithm outperforms the widely used decoder for polar codes (SCL+CRC)
with the same parameters. The performance of the new algorithm for RM codes in
those regimes is in fact close to that of the maximal likelihood decoder.
Finally, the new decoder naturally allows for parallel implementations