2,543 research outputs found
Decoding Reed-Muller codes over product sets
We give a polynomial time algorithm to decode multivariate polynomial codes
of degree up to half their minimum distance, when the evaluation points are
an arbitrary product set , for every . Previously known
algorithms can achieve this only if the set has some very special algebraic
structure, or if the degree is significantly smaller than . We also
give a near-linear time randomized algorithm, which is based on tools from
list-decoding, to decode these codes from nearly half their minimum distance,
provided .
Our result gives an -dimensional generalization of the well known decoding
algorithms for Reed-Solomon codes, and can be viewed as giving an algorithmic
version of the Schwartz-Zippel lemma.Comment: 25 pages, 0 figure
List decoding of a class of affine variety codes
Consider a polynomial in variables and a finite point ensemble . When given the leading monomial of with respect to
a lexicographic ordering we derive improved information on the possible number
of zeros of of multiplicity at least from . We then use this
information to design a list decoding algorithm for a large class of affine
variety codes.Comment: 11 pages, 5 table
Reed-Muller codes for random erasures and errors
This paper studies the parameters for which Reed-Muller (RM) codes over
can correct random erasures and random errors with high probability,
and in particular when can they achieve capacity for these two classical
channels. Necessarily, the paper also studies properties of evaluations of
multi-variate polynomials on random sets of inputs.
For erasures, we prove that RM codes achieve capacity both for very high rate
and very low rate regimes. For errors, we prove that RM codes achieve capacity
for very low rate regimes, and for very high rates, we show that they can
uniquely decode at about square root of the number of errors at capacity.
The proofs of these four results are based on different techniques, which we
find interesting in their own right. In particular, we study the following
questions about , the matrix whose rows are truth tables of all
monomials of degree in variables. What is the most (resp. least)
number of random columns in that define a submatrix having full column
rank (resp. full row rank) with high probability? We obtain tight bounds for
very small (resp. very large) degrees , which we use to show that RM codes
achieve capacity for erasures in these regimes.
Our decoding from random errors follows from the following novel reduction.
For every linear code of sufficiently high rate we construct a new code
, also of very high rate, such that for every subset of coordinates, if
can recover from erasures in , then can recover from errors in .
Specializing this to RM codes and using our results for erasures imply our
result on unique decoding of RM codes at high rate.
Finally, two of our capacity achieving results require tight bounds on the
weight distribution of RM codes. We obtain such bounds extending the recent
\cite{KLP} bounds from constant degree to linear degree polynomials
Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex
Reed-Muller codes are among the most important classes of locally correctable
codes. Currently local decoding of Reed-Muller codes is based on decoding on
lines or quadratic curves to recover one single coordinate. To recover multiple
coordinates simultaneously, the naive way is to repeat the local decoding for
recovery of a single coordinate. This decoding algorithm might be more
expensive, i.e., require higher query complexity. In this paper, we focus on
Reed-Muller codes with usual parameter regime, namely, the total degree of
evaluation polynomials is , where is the code alphabet size
(in fact, can be as big as in our setting). By introducing a novel
variation of codex, i.e., interleaved codex (the concept of codex has been used
for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover
arbitrarily large number of coordinates of a Reed-Muller code
simultaneously at the cost of querying coordinates. It turns out that
our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that
accessing locations is in fact cheaper than repeating the procedure for
accessing a single location for times. Our estimation of success error
probability is based on error probability bound for -wise linearly
independent variables given in \cite{BR94}
Information Sets of Multiplicity Codes
We here provide a method for systematic encoding of the Multiplicity codes
introduced by Kopparty, Saraf and Yekhanin in 2011. The construction is built
on an idea of Kop-party. We properly define information sets for these codes
and give detailed proofs of the validity of Kopparty's construction, that use
generating functions. We also give a complexity estimate of the associated
encoding algorithm.Comment: International Symposium on Information Theory, Jun 2015, Hong-Kong,
China. IEE
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
List Decoding Tensor Products and Interleaved Codes
We design the first efficient algorithms and prove new combinatorial bounds
for list decoding tensor products of codes and interleaved codes. We show that
for {\em every} code, the ratio of its list decoding radius to its minimum
distance stays unchanged under the tensor product operation (rather than
squaring, as one might expect). This gives the first efficient list decoders
and new combinatorial bounds for some natural codes including multivariate
polynomials where the degree in each variable is bounded. We show that for {\em
every} code, its list decoding radius remains unchanged under -wise
interleaving for an integer . This generalizes a recent result of Dinur et
al \cite{DGKS}, who proved such a result for interleaved Hadamard codes
(equivalently, linear transformations). Using the notion of generalized Hamming
weights, we give better list size bounds for {\em both} tensoring and
interleaving of binary linear codes. By analyzing the weight distribution of
these codes, we reduce the task of bounding the list size to bounding the
number of close-by low-rank codewords. For decoding linear transformations,
using rank-reduction together with other ideas, we obtain list size bounds that
are tight over small fields.Comment: 32 page
New Quantum Codes from Evaluation and Matrix-Product Codes
Stabilizer codes obtained via CSS code construction and Steane's enlargement
of subfield-subcodes and matrix-product codes coming from generalized
Reed-Muller, hyperbolic and affine variety codes are studied. Stabilizer codes
with good quantum parameters are supplied, in particular, some binary codes of
lengths 127 and 128 improve the parameters of the codes in
http://www.codetables.de. Moreover, non-binary codes are presented either with
parameters better than or equal to the quantum codes obtained from BCH codes by
La Guardia or with lengths that can not be reached by them
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