171 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
Private Information Retrieval Schemes for Coded Data with Arbitrary Collusion Patterns
In Private Information Retrieval (PIR), one wants to download a file from a
database without revealing to the database which file is being downloaded. Much
attention has been paid to the case of the database being encoded across
several servers, subsets of which can collude to attempt to deduce the
requested file. With the goal of studying the achievable PIR rates in realistic
scenarios, we generalize results for coded data from the case of all subsets of
servers of size colluding, to arbitrary subsets of the servers. We
investigate the effectiveness of previous strategies in this new scenario, and
present new results in the case where the servers are partitioned into disjoint
colluding groups.Comment: Updated with a corrected statement of Theorem
Algebraic Hybrid Satellite-Terrestrial Space-Time Codes for Digital Broadcasting in SFN
Lately, different methods for broadcasting future digital TV in a single
frequency network (SFN) have been under an intensive study. To improve the
transmission to also cover suburban and rural areas, a hybrid scheme may be
used. In hybrid transmission, the signal is transmitted both from a satellite
and from a terrestrial site. In 2008, Y. Nasser et al. proposed to use a double
layer 3D space-time (ST) code in the hybrid 4 x 2 MIMO transmission of digital
TV. In this paper, alternative codes with simpler structure are proposed for
the 4 x 2 hybrid system, and new codes are constructed for the 3 x 2 system.
The performance of the proposed codes is analyzed through computer simulations,
showing a significant improvement over simple repetition schemes. The proposed
codes prove in addition to be very robust in the presence of power imbalance
between the two sites.Comment: 14 pages, 2 figures, submitted to ISIT 201
Fast-Decodable Asymmetric Space-Time Codes from Division Algebras
Multiple-input double-output (MIDO) codes are important in the near-future
wireless communications, where the portable end-user device is physically small
and will typically contain at most two receive antennas. Especially tempting is
the 4 x 2 channel due to its immediate applicability in the digital video
broadcasting (DVB). Such channels optimally employ rate-two space-time (ST)
codes consisting of (4 x 4) matrices. Unfortunately, such codes are in general
very complex to decode, hence setting forth a call for constructions with
reduced complexity.
Recently, some reduced complexity constructions have been proposed, but they
have mainly been based on different ad hoc methods and have resulted in
isolated examples rather than in a more general class of codes. In this paper,
it will be shown that a family of division algebra based MIDO codes will always
result in at least 37.5% worst-case complexity reduction, while maintaining
full diversity and, for the first time, the non-vanishing determinant (NVD)
property. The reduction follows from the fact that, similarly to the Alamouti
code, the codes will be subsets of matrix rings of the Hamiltonian quaternions,
hence allowing simplified decoding. At the moment, such reductions are among
the best known for rate-two MIDO codes. Several explicit constructions are
presented and shown to have excellent performance through computer simulations.Comment: 26 pages, 1 figure, submitted to IEEE Trans. Inf. Theory, October
201
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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