70,500 research outputs found
Estimates on the Size of Symbol Weight Codes
The study of codes for powerlines communication has garnered much interest
over the past decade. Various types of codes such as permutation codes,
frequency permutation arrays, and constant composition codes have been proposed
over the years. In this work we study a type of code called the bounded symbol
weight codes which was first introduced by Versfeld et al. in 2005, and a
related family of codes that we term constant symbol weight codes. We provide
new upper and lower bounds on the size of bounded symbol weight and constant
symbol weight codes. We also give direct and recursive constructions of codes
for certain parameters.Comment: 14 pages, 4 figure
Homological Product Codes
Quantum codes with low-weight stabilizers known as LDPC codes have been
actively studied recently due to their simple syndrome readout circuits and
potential applications in fault-tolerant quantum computing. However, all
families of quantum LDPC codes known to this date suffer from a poor distance
scaling limited by the square-root of the code length. This is in a sharp
contrast with the classical case where good families of LDPC codes are known
that combine constant encoding rate and linear distance. Here we propose the
first family of good quantum codes with low-weight stabilizers. The new codes
have a constant encoding rate, linear distance, and stabilizers acting on at
most qubits, where is the code length. For comparison, all
previously known families of good quantum codes have stabilizers of linear
weight. Our proof combines two techniques: randomized constructions of good
quantum codes and the homological product operation from algebraic topology. We
conjecture that similar methods can produce good stabilizer codes with
stabilizer weight for any . Finally, we apply the homological
product to construct new small codes with low-weight stabilizers.Comment: 49 page
Bounds and Constructions of Singleton-Optimal Locally Repairable Codes with Small Localities
Constructions of optimal locally repairable codes (LRCs) achieving
Singleton-type bound have been exhaustively investigated in recent years. In
this paper, we consider new bounds and constructions of Singleton-optimal LRCs
with minmum distance , locality and minimum distance and
locality , respectively. Firstly, we establish equivalent connections
between the existence of these two families of LRCs and the existence of some
subsets of lines in the projective space with certain properties. Then, we
employ the line-point incidence matrix and Johnson bounds for constant weight
codes to derive new improved bounds on the code length, which are tighter than
known results. Finally, by using some techniques of finite field and finite
geometry, we give some new constructions of Singleton-optimal LRCs, which have
larger length than previous ones
Multiset Combinatorial Batch Codes
Batch codes, first introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai,
mimic a distributed storage of a set of data items on servers, in such
a way that any batch of data items can be retrieved by reading at most some
symbols from each server. Combinatorial batch codes, are replication-based
batch codes in which each server stores a subset of the data items.
In this paper, we propose a generalization of combinatorial batch codes,
called multiset combinatorial batch codes (MCBC), in which data items are
stored in servers, such that any multiset request of items, where any
item is requested at most times, can be retrieved by reading at most
items from each server. The setup of this new family of codes is motivated by
recent work on codes which enable high availability and parallel reads in
distributed storage systems. The main problem under this paradigm is to
minimize the number of items stored in the servers, given the values of
, which is denoted by . We first give a necessary and
sufficient condition for the existence of MCBCs. Then, we present several
bounds on and constructions of MCBCs. In particular, we
determine the value of for any , where
is the maximum size of a binary constant weight code of length
, distance four and weight . We also determine the exact value of
when or
Numerical cubature using error-correcting codes
We present a construction for improving numerical cubature formulas with
equal weights and a convolution structure, in particular equal-weight product
formulas, using linear error-correcting codes. The construction is most
effective in low degree with extended BCH codes. Using it, we obtain several
sequences of explicit, positive, interior cubature formulas with good
asymptotics for each fixed degree as the dimension . Using a
special quadrature formula for the interval [arXiv:math.PR/0408360], we obtain
an equal-weight -cubature formula on the -cube with O(n^{\floor{t/2}})
points, which is within a constant of the Stroud lower bound. We also obtain
-cubature formulas on the -sphere, -ball, and Gaussian with
points when is odd. When is spherically symmetric and
, we obtain points. For each , we also obtain explicit,
positive, interior formulas for the -simplex with points; for
, we obtain O(n) points. These constructions asymptotically improve the
non-constructive Tchakaloff bound.
Some related results were recently found independently by Victoir, who also
noted that the basic construction more directly uses orthogonal arrays.Comment: Dedicated to Wlodzimierz and Krystyna Kuperberg on the occasion of
their 40th anniversary. This version has a major improvement for the n-cub
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