11,192 research outputs found
Subspace Polynomials and Cyclic Subspace Codes
Subspace codes have received an increasing interest recently due to their
application in error-correction for random network coding. In particular,
cyclic subspace codes are possible candidates for large codes with efficient
encoding and decoding algorithms. In this paper we consider such cyclic codes
and provide constructions of optimal codes for which their codewords do not
have full orbits. We further introduce a new way to represent subspace codes by
a class of polynomials called subspace polynomials. We present some
constructions of such codes which are cyclic and analyze their parameters
Coding for Errors and Erasures in Random Network Coding
The problem of error-control in random linear network coding is considered. A
``noncoherent'' or ``channel oblivious'' model is assumed where neither
transmitter nor receiver is assumed to have knowledge of the channel transfer
characteristic. Motivated by the property that linear network coding is
vector-space preserving, information transmission is modelled as the injection
into the network of a basis for a vector space and the collection by the
receiver of a basis for a vector space . A metric on the projective geometry
associated with the packet space is introduced, and it is shown that a minimum
distance decoder for this metric achieves correct decoding if the dimension of
the space is sufficiently large. If the dimension of each codeword
is restricted to a fixed integer, the code forms a subset of a finite-field
Grassmannian, or, equivalently, a subset of the vertices of the corresponding
Grassmann graph. Sphere-packing and sphere-covering bounds as well as a
generalization of the Singleton bound are provided for such codes. Finally, a
Reed-Solomon-like code construction, related to Gabidulin's construction of
maximum rank-distance codes, is described and a Sudan-style ``list-1'' minimum
distance decoding algorithm is provided.Comment: This revised paper contains some minor changes and clarification
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
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