2,397 research outputs found
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
The physics and the mixed Hodge structure of Feynman integrals
This expository text is an invitation to the relation between quantum field
theory Feynman integrals and periods. We first describe the relation between
the Feynman parametrization of loop amplitudes and world-line methods, by
explaining that the first Symanzik polynomial is the determinant of the period
matrix of the graph, and the second Symanzik polynomial is expressed in terms
of world-line Green's functions. We then review the relation between Feynman
graphs and variations of mixed Hodge structures. Finally, we provide an
algorithm for generating the Picard-Fuchs equation satisfied by the all equal
mass banana graphs in a two-dimensional space-time to all loop orders.Comment: v2: 34 pages, 5 figures. Minor changes. References added. String-math
2013 proceeding contributio
Magic state distillation with punctured polar codes
We present a scheme for magic state distillation using punctured polar codes.
Our results build on some recent work by Bardet et al. (ISIT, 2016) who
discovered that polar codes can be described algebraically as decreasing
monomial codes. Using this powerful framework, we construct tri-orthogonal
quantum codes (Bravyi et al., PRA, 2012) that can be used to distill magic
states for the gate. An advantage of these codes is that they permit the
use of the successive cancellation decoder whose time complexity scales as
. We supplement this with numerical simulations for the erasure
channel and dephasing channel. We obtain estimates for the dimensions and error
rates for the resulting codes for block sizes up to for the erasure
channel and for the dephasing channel. The dimension of the
triply-even codes we obtain is shown to scale like for the binary
erasure channel at noise rate and for the dephasing
channel at noise rate . The corresponding bit error rates drop to
roughly for the erasure channel and for
the dephasing channel respectively.Comment: 18 pages, 4 figure
Bhattacharyya parameter of monomials codes for the Binary Erasure Channel: from pointwise to average reliability
Monomial codes were recently equipped with partial order relations, fact that
allowed researchers to discover structural properties and efficient algorithm
for constructing polar codes. Here, we refine the existing order relations in
the particular case of Binary Erasure Channel. The new order relation takes us
closer to the ultimate order relation induced by the pointwise evaluation of
the Bhattacharyya parameter of the synthetic channels. The best we can hope for
is still a partial order relation. To overcome this issue we appeal to related
technique from network theory. Reliability network theory was recently used in
the context of polar coding and more generally in connection with decreasing
monomial codes. In this article, we investigate how the concept of average
reliability is applied for polar codes designed for the binary erasure channel.
Instead of minimizing the error probability of the synthetic channels, for a
particular value of the erasure parameter p, our codes minimize the average
error probability of the synthetic channels. By means of basic network theory
results we determine a closed formula for the average reliability of a
particular synthetic channel, that recently gain the attention of researchers.Comment: 21 pages, 5 figures, 3 tables. Submitted for possible publicatio
Polar codes with a stepped boundary
We consider explicit polar constructions of blocklength
for the two extreme cases of code rates and
For code rates we design codes with complexity order of in code construction, encoding, and decoding. These codes achieve the
vanishing output bit error rates on the binary symmetric channels with any
transition error probability and perform this task with a
substantially smaller redundancy than do other known high-rate codes,
such as BCH codes or Reed-Muller (RM). We then extend our design to the
low-rate codes that achieve the vanishing output error rates with the same
complexity order of and an asymptotically optimal code rate
for the case of Comment: This article has been submitted to ISIT 201
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