156 research outputs found
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
Information Transmission using the Nonlinear Fourier Transform, Part I: Mathematical Tools
The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and
exactly solvable models, is a method for solving integrable partial
differential equations governing wave propagation in certain nonlinear media.
The NFT decorrelates signal degrees-of-freedom in such models, in much the same
way that the Fourier transform does for linear systems. In this three-part
series of papers, this observation is exploited for data transmission over
integrable channels such as optical fibers, where pulse propagation is governed
by the nonlinear Schr\"odinger equation. In this transmission scheme, which can
be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing
commonly used in linear channels, information is encoded in the nonlinear
frequencies and their spectral amplitudes. Unlike most other fiber-optic
transmission schemes, this technique deals with both dispersion and
nonlinearity directly and unconditionally without the need for dispersion or
nonlinearity compensation methods. This first paper explains the mathematical
tools that underlie the method.Comment: This version contains minor updates of IEEE Transactions on
Information Theory, vol. 60, no. 7, pp. 4312--4328, July 201
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