158 research outputs found
A family of optimal locally recoverable codes
A code over a finite alphabet is called locally recoverable (LRC) if every
symbol in the encoding is a function of a small number (at most ) other
symbols. We present a family of LRC codes that attain the maximum possible
value of the distance for a given locality parameter and code cardinality. The
codewords are obtained as evaluations of specially constructed polynomials over
a finite field, and reduce to a Reed-Solomon code if the locality parameter
is set to be equal to the code dimension. The size of the code alphabet for
most parameters is only slightly greater than the code length. The recovery
procedure is performed by polynomial interpolation over points. We also
construct codes with several disjoint recovering sets for every symbol. This
construction enables the system to conduct several independent and simultaneous
recovery processes of a specific symbol by accessing different parts of the
codeword. This property enables high availability of frequently accessed data
("hot data").Comment: Minor changes. This is the final published version of the pape
Constructions of Optimal and Almost Optimal Locally Repairable Codes
Constructions of optimal locally repairable codes (LRCs) in the case of
and over small finite fields were stated as open problems for
LRCs in [I. Tamo \emph{et al.}, "Optimal locally repairable codes and
connections to matroid theory", \emph{2013 IEEE ISIT}]. In this paper, these
problems are studied by constructing almost optimal linear LRCs, which are
proven to be optimal for certain parameters, including cases for which . More precisely, linear codes for given length, dimension, and
all-symbol locality are constructed with almost optimal minimum distance.
`Almost optimal' refers to the fact that their minimum distance differs by at
most one from the optimal value given by a known bound for LRCs. In addition to
these linear LRCs, optimal LRCs which do not require a large field are
constructed for certain classes of parameters.Comment: 5 pages, conferenc
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
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
Capacity of Locally Recoverable Codes
Motivated by applications in distributed storage, the notion of a locally
recoverable code (LRC) was introduced a few years back. In an LRC, any
coordinate of a codeword is recoverable by accessing only a small number of
other coordinates. While different properties of LRCs have been well-studied,
their performance on channels with random erasures or errors has been mostly
unexplored. In this note, we analyze the performance of LRCs over such
stochastic channels. In particular, for input-symmetric discrete memoryless
channels, we give a tight characterization of the gap to Shannon capacity when
LRCs are used over the channel.Comment: Invited paper to the Information Theory Workshop (ITW) 201
Coding with Constraints: Minimum Distance Bounds and Systematic Constructions
We examine an error-correcting coding framework in which each coded symbol is
constrained to be a function of a fixed subset of the message symbols. With an
eye toward distributed storage applications, we seek to design systematic codes
with good minimum distance that can be decoded efficiently. On this note, we
provide theoretical bounds on the minimum distance of such a code based on the
coded symbol constraints. We refine these bounds in the case where we demand a
systematic linear code. Finally, we provide conditions under which each of
these bounds can be achieved by choosing our code to be a subcode of a
Reed-Solomon code, allowing for efficient decoding. This problem has been
considered in multisource multicast network error correction. The problem setup
is also reminiscent of locally repairable codes.Comment: Submitted to ISIT 201
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