31,455 research outputs found
List and Unique Error-Erasure Decoding of Interleaved Gabidulin Codes with Interpolation Techniques
A new interpolation-based decoding principle for interleaved Gabidulin codes
is presented. The approach consists of two steps: First, a multi-variate
linearized polynomial is constructed which interpolates the coefficients of the
received word and second, the roots of this polynomial have to be found. Due to
the specific structure of the interpolation polynomial, both steps
(interpolation and root-finding) can be accomplished by solving a linear system
of equations. This decoding principle can be applied as a list decoding
algorithm (where the list size is not necessarily bounded polynomially) as well
as an efficient probabilistic unique decoding algorithm. For the unique
decoder, we show a connection to known unique decoding approaches and give an
upper bound on the failure probability. Finally, we generalize our approach to
incorporate not only errors, but also row and column erasures.Comment: accepted for Designs, Codes and Cryptography; presented in part at
WCC 2013, Bergen, Norwa
On the Geometry of Balls in the Grassmannian and List Decoding of Lifted Gabidulin Codes
The finite Grassmannian is defined as the set of all
-dimensional subspaces of the ambient space . Subsets of
the finite Grassmannian are called constant dimension codes and have recently
found an application in random network coding. In this setting codewords from
are sent through a network channel and, since errors may
occur during transmission, the received words can possible lie in
, where . In this paper, we study the balls in
with center that is not necessarily in
. We describe the balls with respect to two different
metrics, namely the subspace and the injection metric. Moreover, we use two
different techniques for describing these balls, one is the Pl\"ucker embedding
of , and the second one is a rational parametrization of
the matrix representation of the codewords.
With these results, we consider the problem of list decoding a certain family
of constant dimension codes, called lifted Gabidulin codes. We describe a way
of representing these codes by linear equations in either the matrix
representation or a subset of the Pl\"ucker coordinates. The union of these
equations and the equations which arise from the description of the ball of a
given radius in the Grassmannian describe the list of codewords with distance
less than or equal to the given radius from the received word.Comment: To be published in Designs, Codes and Cryptography (Springer
A Repair Framework for Scalar MDS Codes
Several works have developed vector-linear maximum-distance separable (MDS)
storage codes that min- imize the total communication cost required to repair a
single coded symbol after an erasure, referred to as repair bandwidth (BW).
Vector codes allow communicating fewer sub-symbols per node, instead of the
entire content. This allows non trivial savings in repair BW. In sharp
contrast, classic codes, like Reed- Solomon (RS), used in current storage
systems, are deemed to suffer from naive repair, i.e. downloading the entire
stored message to repair one failed node. This mainly happens because they are
scalar-linear. In this work, we present a simple framework that treats scalar
codes as vector-linear. In some cases, this allows significant savings in
repair BW. We show that vectorized scalar codes exhibit properties that
simplify the design of repair schemes. Our framework can be seen as a finite
field analogue of real interference alignment. Using our simplified framework,
we design a scheme that we call clique-repair which provably identifies the
best linear repair strategy for any scalar 2-parity MDS code, under some
conditions on the sub-field chosen for vectorization. We specify optimal repair
schemes for specific (5,3)- and (6,4)-Reed- Solomon (RS) codes. Further, we
present a repair strategy for the RS code currently deployed in the Facebook
Analytics Hadoop cluster that leads to 20% of repair BW savings over naive
repair which is the repair scheme currently used for this code.Comment: 10 Pages; accepted to IEEE JSAC -Distributed Storage 201
A Complete Characterization of Irreducible Cyclic Orbit Codes and their Pl\"ucker Embedding
Constant dimension codes are subsets of the finite Grassmann variety. The
study of these codes is a central topic in random linear network coding theory.
Orbit codes represent a subclass of constant dimension codes. They are defined
as orbits of a subgroup of the general linear group on the Grassmannian. This
paper gives a complete characterization of orbit codes that are generated by an
irreducible cyclic group, i.e. a group having one generator that has no
non-trivial invariant subspace. We show how some of the basic properties of
these codes, the cardinality and the minimum distance, can be derived using the
isomorphism of the vector space and the extension field. Furthermore, we
investigate the Pl\"ucker embedding of these codes and show how the orbit
structure is preserved in the embedding.Comment: submitted to Designs, Codes and Cryptograph
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