1,365 research outputs found
Upper bounds on maximum lengths of Singleton-optimal locally repairable codes
A locally repairable code is called Singleton-optimal if it achieves the
Singleton-type bound. Such codes are of great theoretic interest in the study
of locally repairable codes. In the recent years there has been a great amount
of work on this topic. One of the main problems in this topic is to determine
the largest length of a q-ary Singleton-optimal locally repairable code for
given locality and minimum distance. Unlike classical MDS codes, the maximum
length of Singleton? Optimal locally repairable codes are very sensitive to
minimum distance and locality. Thus, it is more challenging and complicated to
investigate the maximum length of Singleton-optimal locally repairable codes.
In literature, there has been already some research on this problem. However,
most of work is concerned with some specific parameter regime such as small
minimum distance and locality, and rely on the constraint that (r + 1)|n and
recovery sets are disjoint, where r is locality and n is the code length. In
this paper we study the problem for large range of parameters including the
case where minimum distance is proportional to length. In addition, we also
derive some upper bounds on the maximum length of Singleton-optimal locally
repairable codes with small minimum distance by removing this constraint. It
turns out that even without the constraint we still get better upper bounds for
codes with small locality and distance compared with known results.
Furthermore, based on our upper bounds for codes with small distance and
locality and some propagation rule that we propose in this paper, we are able
to derive some upper bounds for codes with relatively large distance and
locality assuming that (r + 1)|n and recovery sets are disjoint
Optimal Locally Repairable and Secure Codes for Distributed Storage Systems
This paper aims to go beyond resilience into the study of security and
local-repairability for distributed storage systems (DSS). Security and
local-repairability are both important as features of an efficient storage
system, and this paper aims to understand the trade-offs between resilience,
security, and local-repairability in these systems. In particular, this paper
first investigates security in the presence of colluding eavesdroppers, where
eavesdroppers are assumed to work together in decoding stored information.
Second, the paper focuses on coding schemes that enable optimal local repairs.
It further brings these two concepts together, to develop locally repairable
coding schemes for DSS that are secure against eavesdroppers.
The main results of this paper include: a. An improved bound on the secrecy
capacity for minimum storage regenerating codes, b. secure coding schemes that
achieve the bound for some special cases, c. a new bound on minimum distance
for locally repairable codes, d. code construction for locally repairable codes
that attain the minimum distance bound, and e. repair-bandwidth-efficient
locally repairable codes with and without security constraints.Comment: Submitted to IEEE Transactions on Information Theor
On the Combinatorics of Locally Repairable Codes via Matroid Theory
This paper provides a link between matroid theory and locally repairable
codes (LRCs) that are either linear or more generally almost affine. Using this
link, new results on both LRCs and matroid theory are derived. The parameters
of LRCs are generalized to matroids, and the matroid
analogue of the generalized Singleton bound in [P. Gopalan et al., "On the
locality of codeword symbols," IEEE Trans. Inf. Theory] for linear LRCs is
given for matroids. It is shown that the given bound is not tight for certain
classes of parameters, implying a nonexistence result for the corresponding
locally repairable almost affine codes, that are coined perfect in this paper.
Constructions of classes of matroids with a large span of the parameters
and the corresponding local repair sets are given. Using
these matroid constructions, new LRCs are constructed with prescribed
parameters. The existence results on linear LRCs and the nonexistence results
on almost affine LRCs given in this paper strengthen the nonexistence and
existence results on perfect linear LRCs given in [W. Song et al., "Optimal
locally repairable codes," IEEE J. Sel. Areas Comm.].Comment: 48 pages. Submitted for publication. In this version: The text has
been edited to improve the readability. Parameter d for matroids is now
defined by the use of the rank function instead of the dual matroid. Typos
are corrected. Section III is divided into two parts, and some numberings of
theorems etc. have been change
Security in Locally Repairable Storage
In this paper we extend the notion of {\em locally repairable} codes to {\em
secret sharing} schemes. The main problem that we consider is to find optimal
ways to distribute shares of a secret among a set of storage-nodes
(participants) such that the content of each node (share) can be recovered by
using contents of only few other nodes, and at the same time the secret can be
reconstructed by only some allowable subsets of nodes. As a special case, an
eavesdropper observing some set of specific nodes (such as less than certain
number of nodes) does not get any information. In other words, we propose to
study a locally repairable distributed storage system that is secure against a
{\em passive eavesdropper} that can observe some subsets of nodes.
We provide a number of results related to such systems including upper-bounds
and achievability results on the number of bits that can be securely stored
with these constraints.Comment: This paper has been accepted for publication in IEEE Transactions of
Information Theor
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
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
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