273 research outputs found
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 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
Codes with Locality for Two Erasures
In this paper, we study codes with locality that can recover from two
erasures via a sequence of two local, parity-check computations. By a local
parity-check computation, we mean recovery via a single parity-check equation
associated to small Hamming weight. Earlier approaches considered recovery in
parallel; the sequential approach allows us to potentially construct codes with
improved minimum distance. These codes, which we refer to as locally
2-reconstructible codes, are a natural generalization along one direction, of
codes with all-symbol locality introduced by Gopalan \textit{et al}, in which
recovery from a single erasure is considered. By studying the Generalized
Hamming Weights of the dual code, we derive upper bounds on the minimum
distance of locally 2-reconstructible codes and provide constructions for a
family of codes based on Tur\'an graphs, that are optimal with respect to this
bound. The minimum distance bound derived here is universal in the sense that
no code which permits all-symbol local recovery from erasures can have
larger minimum distance regardless of approach adopted. Our approach also leads
to a new bound on the minimum distance of codes with all-symbol locality for
the single-erasure case.Comment: 14 pages, 3 figures, Updated for improved readabilit
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
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