5,156 research outputs found
Parameters of AG codes from vector bundles
AbstractWe investigate the parameters of the algebraic–geometric codes constructed from vector bundles on a projective variety defined over a finite field. In the case of curves we give a method of constructing weakly stable bundles using restriction of vector bundles on algebraic surfaces and illustrate the result by some examples
AG Codes from Polyhedral Divisors
A description of complete normal varieties with lower dimensional torus
action has been given by Altmann, Hausen, and Suess, generalizing the theory of
toric varieties. Considering the case where the acting torus T has codimension
one, we describe T-invariant Weil and Cartier divisors and provide formulae for
calculating global sections, intersection numbers, and Euler characteristics.
As an application, we use divisors on these so-called T-varieties to define new
evaluation codes called T-codes. We find estimates on their minimum distance
using intersection theory. This generalizes the theory of toric codes and
combines it with AG codes on curves. As the simplest application of our general
techniques we look at codes on ruled surfaces coming from decomposable vector
bundles. Already this construction gives codes that are better than the related
product code. Further examples show that we can improve these codes by
constructing more sophisticated T-varieties. These results suggest to look
further for good codes on T-varieties.Comment: 30 pages, 9 figures; v2: replaced fansy cycles with fansy divisor
Wolf Barth (1942--2016)
In this article we describe the life and work of Wolf Barth who died on 30th
December 2016. Wolf Barth's contributions to algebraic variety span a wide
range of subjects. His achievements range from what is now called the
Barth-Lefschetz theorems to his fundamental contributions to the theory of
algebraic surfaces and moduli of vector bundles, and include his later work on
algebraic surfaces with many singularities, culminating in the famous Barth
sextic.Comment: accepted for publication in Jahresbericht der Deutschen
Mathematiker-Vereinigung, obituary, 17 pages, 2 figures, 1 phot
Subspace code constructions
We improve on the lower bound of the maximum number of planes of mutually intersecting in at most one point leading to the following
lower bound: for
constant dimension subspace codes. We also construct two new non-equivalent
constant dimension subspace orbit-codes
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