637 research outputs found
Optimal Linear and Cyclic Locally Repairable Codes over Small Fields
We consider locally repairable codes over small fields and propose
constructions of optimal cyclic and linear codes in terms of the dimension for
a given distance and length. Four new constructions of optimal linear codes
over small fields with locality properties are developed. The first two
approaches give binary cyclic codes with locality two. While the first
construction has availability one, the second binary code is characterized by
multiple available repair sets based on a binary Simplex code. The third
approach extends the first one to q-ary cyclic codes including (binary)
extension fields, where the locality property is determined by the properties
of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear
codes with locality greater than two are obtained by the fourth construction.Comment: IEEE Information Theory Workshop (ITW) 2015, Apr 2015, Jerusalem,
Israe
On Binary Matroid Minors and Applications to Data Storage over Small Fields
Locally repairable codes for distributed storage systems have gained a lot of
interest recently, and various constructions can be found in the literature.
However, most of the constructions result in either large field sizes and hence
too high computational complexity for practical implementation, or in low rates
translating into waste of the available storage space. In this paper we address
this issue by developing theory towards code existence and design over a given
field. This is done via exploiting recently established connections between
linear locally repairable codes and matroids, and using matroid-theoretic
characterisations of linearity over small fields. In particular, nonexistence
can be shown by finding certain forbidden uniform minors within the lattice of
cyclic flats. It is shown that the lattice of cyclic flats of binary matroids
have additional structure that significantly restricts the possible locality
properties of -linear storage codes. Moreover, a collection of
criteria for detecting uniform minors from the lattice of cyclic flats of a
given matroid is given, which is interesting in its own right.Comment: 14 pages, 2 figure
On Distributed Storage Codes
Distributed storage systems are studied. The interest in such system has become relatively wide due to the increasing amount of information needed to be stored in data centers or different kinds of cloud systems. There are many kinds of solutions for storing the information into distributed devices regarding the needs of the system designer. This thesis studies the questions of designing such storage systems and also fundamental limits of such systems. Namely, the subjects of interest of this thesis include heterogeneous distributed storage systems, distributed storage systems with the exact repair property, and locally repairable codes. For distributed storage systems with either functional or exact repair, capacity results are proved. In the case of locally repairable codes, the minimum distance is studied.
Constructions for exact-repairing codes between minimum bandwidth regeneration (MBR) and minimum storage regeneration (MSR) points are given. These codes exceed the time-sharing line of the extremal points in many cases. Other properties of exact-regenerating codes are also studied. For the heterogeneous setup, the main result is that the capacity of such systems is always smaller than or equal to the capacity of a homogeneous system with symmetric repair with average node size and average repair bandwidth. A randomized construction for a locally repairable code with good minimum distance is given. It is shown that a random linear code of certain natural type has a good minimum distance with high probability. Other properties of locally repairable codes are also studied.Siirretty Doriast
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
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
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