Mathematics Commons^™

Codes Over Rings From Curves Of Higher Genus, José Felipe Voloch, Judy L. Walker

Department of Mathematics: Faculty Publications

We construct certain error-correcting codes over finite rings and estimate their parameters. These codes are constructed using plane curves and the estimates for their parameters rely on constructing “lifts” of these curves and then estimating the size of certain exponential sums.

THE purpose of this paper is to construct certain error-correcting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools; notably, an estimate for the dimension of trace codes over rings (generalizing work of van der Vlugt over fields and some results on lifts of affin curves from field of characteristic p …

Lack Of Time-Delay Robustness For Stabilization Of A Structural Acoustics Model, George Avalos, Irena Lasiecka, Richard Rebarber

Department of Mathematics: Faculty Publications

In this paper we consider a natural robustness question for a model for structural acoustics. This model, which has been of great interest in recent years, is represented by a wave equation in R^2 coupled to a Kelvin--Voigt beam; the coupling is natural physically, and is represented mathematically by highly unbounded operators. We assume that the observation consists of point evaluation of the beam position, the beam velocity, and the wave velocity. We are interested in the effect of arbitrarily small delays in the feedback loop on a controller that uses these observations. We show that it is not possible …

Reconstructing Subsets Of Reals, A. J. Radcliffe, A. D. Scott

Department of Mathematics: Faculty Publications

We consider the problem of reconstructing a set of real numbers up to translation from the multiset of its subsets of fixed size, given up to translation. This is impossible in general: for instance almost all subsets of Z contain infinitely many translates of every finite subset of Z. We therefore restrict our attention to subsets of R which are locally finite; those which contain only finitely many translates of any given finite set of size at least 2. We prove that every locally finite subset of R is reconstructible from the multiset of its 3-subsets, given up to …

Algebraic Geometric Codes Over Rings, Judy L. Walker

Department of Mathematics: Faculty Publications

The techniques of algebraic geometry have been widely and successfully applied to the study of linear codes over finite fields since the early 1980’s. Recently, there has been an increased interest in the study of linear codes over finite rings. In this paper, we combine these two approaches to coding theory by introducing the study of algebraic geometric codes over rings. In addition to defining these new codes, we prove several results about their properties.