803 research outputs found
Higher Hamming weights for locally recoverable codes on algebraic curves
We study the locally recoverable codes on algebraic curves. In the first part
of this article, we provide a bound of generalized Hamming weight of these
codes. Whereas in the second part, we propose a new family of algebraic
geometric LRC codes, that are LRC codes from Norm-Trace curve. Finally, using
some properties of Hermitian codes, we improve the bounds of distance proposed
in [1] for some Hermitian LRC codes.
[1] A. Barg, I. Tamo, and S. Vlladut. Locally recoverable codes on algebraic
curves. arXiv preprint arXiv:1501.04904, 2015
Locally recoverable codes on surfaces
A linear error correcting code is a subspace of a finite-dimensional space
over a finite field with a fixed coordinate system. Such a code is said to be
locally recoverable with locality if, for every coordinate, its value at a
codeword can be deduced from the value of (certain) other coordinates of
the codeword. These codes have found many recent applications, e.g., to
distributed cloud storage. We will discuss the problem of constructing good
locally recoverable codes and present some constructions using algebraic
surfaces that improve previous constructions and sometimes provide codes that
are optimal in a precise sense. The main conceptual contribution of this paper
is to consider surfaces fibered over a curve in such a way that each recovery
set is constructed from points in a single fiber. This allows us to use the
geometry of the fiber to guarantee the local recoverability and use the global
geometry of the surface to get a hold on the standard parameters of our codes.
We look in detail at situations where the fibers are rational or elliptic
curves and provide many examples applying our methods.Comment: Revised version; incorporates suggestions by referee
Optimal locally repairable codes of distance and via cyclic codes
Like classical block codes, a locally repairable code also obeys the
Singleton-type bound (we call a locally repairable code {\it optimal} if it
achieves the Singleton-type bound). In the breakthrough work of \cite{TB14},
several classes of optimal locally repairable codes were constructed via
subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in
\cite{TB14} are upper bounded by the code alphabet size . Recently, it was
proved through extension of construction in \cite{TB14} that length of -ary
optimal locally repairable codes can be in \cite{JMX17}. Surprisingly,
\cite{BHHMV16} presented a few examples of -ary optimal locally repairable
codes of small distance and locality with code length achieving roughly .
Very recently, it was further shown in \cite{LMX17} that there exist -ary
optimal locally repairable codes with length bigger than and distance
propositional to .
Thus, it becomes an interesting and challenging problem to construct new
families of -ary optimal locally repairable codes of length bigger than
.
In this paper, we construct a class of optimal locally repairable codes of
distance and with unbounded length (i.e., length of the codes is
independent of the code alphabet size). Our technique is through cyclic codes
with particular generator and parity-check polynomials that are carefully
chosen
- …