220 research outputs found
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
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
Error Correcting Codes on Algebraic Surfaces
Error correcting codes are defined and important parameters for a code are
explained. Parameters of new codes constructed on algebraic surfaces are
studied. In particular, codes resulting from blowing up points in \proj^2 are
briefly studied, then codes resulting from ruled surfaces are covered. Codes
resulting from ruled surfaces over curves of genus 0 are completely analyzed,
and some codes are discovered that are better than direct product Reed Solomon
codes of similar length. Ruled surfaces over genus 1 curves are also studied,
but not all classes are completely analyzed. However, in this case a family of
codes are found that are comparable in performance to the direct product code
of a Reed Solomon code and a Goppa code. Some further work is done on surfaces
from higher genus curves, but there remains much work to be done in this
direction to understand fully the resulting codes. Codes resulting from blowing
points on surfaces are also studied, obtaining necessary parameters for
constructing infinite families of such codes.
Also included is a paper giving explicit formulas for curves with more
\field{q}-rational points than were previously known for certain combinations
of field size and genus. Some upper bounds are now known to be optimal from
these examples.Comment: This is Chris Lomont's PhD thesis about error correcting codes from
algebriac surface
On Frobenius incidence varieties of linear subspaces over finite fields
We define Frobenius incidence varieties by means of the incidence relation of
Frobenius images of linear subspaces in a fixed vector space over a finite
field, and investigate their properties such as supersingularity, Betti numbers
and unirationality. These varieties are variants of the Deligne-Lusztig
varieties. We then study the lattices associated with algebraic cycles on them.
We obtain a positive-definite lattice of rank 84 that yields a dense sphere
packing from a 4-dimensional Frobenius incidence variety in characteristic 2.Comment: 24 pages, no figures; Introduction is changed. New references are
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Toric surface codes and Minkowski sums
Toric codes are evaluation codes obtained from an integral convex polytope and finite field \F_q. They are, in a sense, a natural
extension of Reed-Solomon codes, and have been studied recently by J. Hansen
and D. Joyner. In this paper, we obtain upper and lower bounds on the minimum
distance of a toric code constructed from a polygon by
examining Minkowski sum decompositions of subpolygons of . Our results give
a simple and unifying explanation of bounds of Hansen and empirical results of
Joyner; they also apply to previously unknown cases.Comment: 15 pages, 7 figures; This version contains some minor editorial
revisions -- to appear SIAM Journal on Discrete Mathematic
Codes as fractals and noncommutative spaces
We consider the CSS algorithm relating self-orthogonal classical linear codes
to q-ary quantum stabilizer codes and we show that to such a pair of a
classical and a quantum code one can associate geometric spaces constructed
using methods from noncommutative geometry, arising from rational
noncommutative tori and finite abelian group actions on Cuntz algebras and
fractals associated to the classical codes.Comment: 18 pages LaTeX, one png figur
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