3,271 research outputs found
A family of optimal locally recoverable codes
A code over a finite alphabet is called locally recoverable (LRC) if every
symbol in the encoding is a function of a small number (at most ) other
symbols. We present a family of LRC codes that attain the maximum possible
value of the distance for a given locality parameter and code cardinality. The
codewords are obtained as evaluations of specially constructed polynomials over
a finite field, and reduce to a Reed-Solomon code if the locality parameter
is set to be equal to the code dimension. The size of the code alphabet for
most parameters is only slightly greater than the code length. The recovery
procedure is performed by polynomial interpolation over points. We also
construct codes with several disjoint recovering sets for every symbol. This
construction enables the system to conduct several independent and simultaneous
recovery processes of a specific symbol by accessing different parts of the
codeword. This property enables high availability of frequently accessed data
("hot data").Comment: Minor changes. This is the final published version of the pape
Asymptotically optimal cooperative wireless networks with reduced signaling complexity
This paper considers an orthogonal amplify-and-forward (OAF) protocol for cooperative relay communication over Rayleigh-fading channels in which the intermediate relays are permitted to linearly transform the received signal and where the source and relays transmit for equal time durations. The diversity-multiplexing gain (D-MG) tradeoff of the equivalent space-time channel associated to this protocol is determined and a cyclic-division-algebra-based D-MG optimal code constructed. The transmission or signaling alphabet of this code is the union of the QAM constellation and a rotated version of QAM. The size of this signaling alphabet is small in comparison with prior D-MG optimal constructions in the literature and is independent of the number of participating nodes in the network
Asymptotically good binary linear codes with asymptotically good self-intersection spans
If C is a binary linear code, let C^2 be the linear code spanned by
intersections of pairs of codewords of C. We construct an asymptotically good
family of binary linear codes such that, for C ranging in this family, the C^2
also form an asymptotically good family. For this we use algebraic-geometry
codes, concatenation, and a fair amount of bilinear algebra.
More precisely, the two main ingredients used in our construction are, first,
a description of the symmetric square of an odd degree extension field in terms
only of field operations of small degree, and second, a recent result of
Garcia-Stichtenoth-Bassa-Beelen on the number of points of curves on such an
odd degree extension field.Comment: 18 pages; v2->v3: expanded introduction and bibliography + various
minor change
Polynomial time algorithms for multicast network code construction
The famous max-flow min-cut theorem states that a source node s can send information through a network (V, E) to a sink node t at a rate determined by the min-cut separating s and t. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures
Multilevel Polarization of Polar Codes Over Arbitrary Discrete Memoryless Channels
It is shown that polar codes achieve the symmetric capacity of discrete
memoryless channels with arbitrary input alphabet sizes. It is shown that in
general, channel polarization happens in several, rather than only two levels
so that the synthesized channels are either useless, perfect or "partially
perfect". Any subset of the channel input alphabet which is closed under
addition, induces a coset partition of the alphabet through its shifts. For any
such partition of the input alphabet, there exists a corresponding partially
perfect channel whose outputs uniquely determine the coset to which the channel
input belongs. By a slight modification of the encoding and decoding rules, it
is shown that perfect transmission of certain information symbols over
partially perfect channels is possible. Our result is general regarding both
the cardinality and the algebraic structure of the channel input alphabet; i.e
we show that for any channel input alphabet size and any Abelian group
structure on the alphabet, polar codes are optimal. It is also shown through an
example that polar codes when considered as group/coset codes, do not achieve
the capacity achievable using coset codes over arbitrary channels
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
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