Systems theoretic methods in decoding

In this paper we show how ideas based on system theoretic modeling of linear behaviors may be used for decoding of linear codes. In particular we show how Sudan's bivariate interpolation approach to list decoding of RS codes allows a system theoretic interpretation

Interaction between current imbalance and magnetization in LHC cables

The quality of the magnetic field in superconducting accelerator magnets is associated with the properties of the superconducting cable. Current imbalances due to coupling currents ¿I, as large as 100 A, are induced by spatial variations of the field sweep rate and contact resistances. During injection at a constant field all magnetic field components show a decay behavior. The decay is caused by a diffusion of coupling currents into the whole magnet. This results in a redistribution of the transport current among the strands and causes a demagnetization of the superconducting cable. As soon as the field is ramped up again after the end of injection, the magnetization rapidly recovers from the decay and follows the course of the original hysteresis curve. In order to clarify the interactions between the changes in current and magnetization during injection the authors performed a number of experiments. A magnetic field with a spatially periodic pattern was applied to a superconducting wire in order to simulate the coupling behavior in a magnet. This model system was placed into a stand for magnetization measurements and the influence of different powering conditions was analyze

An iterative algorithm for parametrization of shortest length shift registers over finite rings

The construction of shortest feedback shift registers for a finite sequence S_1,...,S_N is considered over the finite ring Z_{p^r}. A novel algorithm is presented that yields a parametrization of all shortest feedback shift registers for the sequence of numbers S_1,...,S_N, thus solving an open problem in the literature. The algorithm iteratively processes each number, starting with S_1, and constructs at each step a particular type of minimal Gr\"obner basis. The construction involves a simple update rule at each step which leads to computational efficiency. It is shown that the algorithm simultaneously computes a similar parametrization for the reciprocal sequence S_N,...,S_1.Comment: Submitte

On minimality of convolutional ring encoders

Convolutional codes are considered with code sequences modeled as semi-infinite Laurent series. It is well known that a convolutional code C over a finite group G has a minimal trellis representation that can be derived from code sequences. It is also well known that, for the case that G is a finite field, any polynomial encoder of C can be algebraically manipulated to yield a minimal polynomial encoder whose controller canonical realization is a minimal trellis. In this paper we seek to extend this result to the finite ring case G = ℤ_{p^r} by introducing a so-called "p-encoder". We show how to manipulate a polynomial encoding scheme of a noncatastrophic convolutional code over ℤ_{p^r} to produce a particular type of p-encoder ("minimal p-encoder") whose controller canonical realization is a minimal trellis with nonlinear features. The minimum number of trellis states is then expressed as p^γ, where γ is the sum of the row degrees of the minimal p-encoder. In particular, we show that any convolutional code over ℤ_{p^r} admits a delay-free p-encoder which implies the novel result that delay-freeness is not a property of the code but of the encoder, just as in the field case. We conjecture that a similar result holds with respect to catastrophicity, i.e., any catastrophic convolutional code over ℤ_{p^r} admits a noncatastrophic p-encoder. © 2009 IEEE

Realization and partial fractions

AbstractWe discuss the relation between two intrinsically different proposals that have been made in the literature concerning the representation by constant matrices of rational matrices given in fractional form. It turns out that the relation is most naturally studied in the framework of partial-fraction decompositions. We develop the realization theory for decompositions with respect to arbitrary complementary parts of the extended complex plane which may, for instance, correspond to stability and instability. An isomorphism is obtained which connects the spaces used in the two methods, and several identities relating to the McMillan degree are derived in a direct way. Finally, a new computational procedure is given to obtain the partial-fraction decomposition of a rational matrix given in fractional form