384 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
Bounds on Binary Locally Repairable Codes Tolerating Multiple Erasures
Recently, locally repairable codes has gained significant interest for their
potential applications in distributed storage systems. However, most
constructions in existence are over fields with size that grows with the number
of servers, which makes the systems computationally expensive and difficult to
maintain. Here, we study linear locally repairable codes over the binary field,
tolerating multiple local erasures. We derive bounds on the minimum distance on
such codes, and give examples of LRCs achieving these bounds. Our main
technical tools come from matroid theory, and as a byproduct of our proofs, we
show that the lattice of cyclic flats of a simple binary matroid is atomic.Comment: 9 pages, 1 figure. Parts of this paper were presented at IZS 2018.
This extended arxiv version includes corrected versions of Theorem 1.4 and
Proposition 6 that appeared in the IZS 2018 proceeding
Optimal Binary Locally Repairable Codes via Anticodes
This paper presents a construction for several families of optimal binary
locally repairable codes (LRCs) with small locality (2 and 3). This
construction is based on various anticodes. It provides binary LRCs which
attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal
with respect to the Griesmer bound
Local Codes with Addition Based Repair
We consider the complexities of repair algorithms for locally repairable
codes and propose a class of codes that repair single node failures using
addition operations only, or codes with addition based repair. We construct two
families of codes with addition based repair. The first family attains distance
one less than the Singleton-like upper bound, while the second family attains
the Singleton-like upper bound
- …