36 research outputs found
Gr\"obner Bases for Linearized Polynomials
In this work we develop the theory of Gr\"obner bases for modules over the
ring of univariate linearized polynomials with coefficients from a finite
field
Iterative List-Decoding of Gabidulin Codes via Gr\"obner Based Interpolation
We show how Gabidulin codes can be list decoded by using an iterative
parametrization approach. For a given received word, our decoding algorithm
processes its entries one by one, constructing four polynomials at each step.
This then yields a parametrization of interpolating solutions for the data so
far. From the final result a list of all codewords that are closest to the
received word with respect to the rank metric is obtained.Comment: Submitted to IEEE Information Theory Workshop 2014 in Hobart,
Australi
List-Decoding Gabidulin Codes via Interpolation and the Euclidean Algorithm
We show how Gabidulin codes can be list decoded by using a parametrization
approach. For this we consider a certain module in the ring of linearized
polynomials and find a minimal basis for this module using the Euclidean
algorithm with respect to composition of polynomials. For a given received
word, our decoding algorithm computes a list of all codewords that are closest
to the received word with respect to the rank metric.Comment: Submitted to ISITA 2014, IEICE copyright upon acceptanc
Reed-Solomon list decoding from a system-theoretic perspective
In this paper, the Sudan-Guruswami approach to list decoding of Reed-Solomon (RS) codes is cast in a system-theoretic framework. With the data, a set of trajectories or time series is associated which is then modeled as a so-called behavior. In this way, a connection is made with the behavioral approach to system theory. It is shown how a polynomial representation of the modeling behavior gives rise to the bivariate interpolating polynomials of the Sudan-Guruswami approach. The concept of "weighted row reduced" is introduced and used to achieve minimality. Two decoding methods are derived and a parametrization of all bivariate interpolating polynomials is given
Row reduced representations of behaviors over finite rings
Row reduced representations of behaviors over fields posses a number of useful properties. Perhaps the most important feature is the predictable degree property. This property allows a finite parametrization of the module generated by the rows of the row reduced matrix with prior computable bounds. In this paper we study row-reducedness of representations of behaviors over rings of the form , where is a prime number. Using a restricted calculus within we derive a meaningful and computable notion of row-reducedness
A Multi-Observer Based Estimation Framework for Nonlinear Systems under Sensor Attacks
We address the problem of state estimation and attack isolation for general
discrete-time nonlinear systems when sensors are corrupted by (potentially
unbounded) attack signals. For a large class of nonlinear plants and observers,
we provide a general estimation scheme, built around the idea of sensor
redundancy and multi-observer, capable of reconstructing the system state in
spite of sensor attacks and noise. This scheme has been proposed by others for
linear systems/observers and here we propose a unifying framework for a much
larger class of nonlinear systems/observers. Using the proposed estimator, we
provide an isolation algorithm to pinpoint attacks on sensors during sliding
time windows. Simulation results are presented to illustrate the performance of
our tools.Comment: arXiv admin note: text overlap with arXiv:1806.0648