415 research outputs found
Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory
We consider the question of determining the maximum number of
-rational points that can lie on a hypersurface of a given degree
in a weighted projective space over the finite field , or in
other words, the maximum number of zeros that a weighted homogeneous polynomial
of a given degree can have in the corresponding weighted projective space over
. In the case of classical projective spaces, this question has
been answered by J.-P. Serre. In the case of weighted projective spaces, we
give some conjectures and partial results. Applications to coding theory are
included and an appendix providing a brief compendium of results about weighted
projective spaces is also included
Regularity and algebraic properties of certain lattice ideals
We study the regularity and the algebraic properties of certain lattice
ideals. We establish a map I --> I\~ between the family of graded lattice
ideals in an N-graded polynomial ring over a field K and the family of graded
lattice ideals in a polynomial ring with the standard grading. This map is
shown to preserve the complete intersection property and the regularity of I
but not the degree. We relate the Hilbert series and the generators of I and
I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the
regularity of certain lattice ideals is additive in a certain sense. Then, we
give some applications. For finite fields, we give a formula for the regularity
of the vanishing ideal of a degenerate torus in terms of the Frobenius number
of a semigroup. We construct vanishing ideals, over finite fields, with
prescribed regularity and degree of a certain type. Let X be a subset of a
projective space over a field K. It is shown that the vanishing ideal of X is a
lattice ideal of dimension 1 if and only if X is a finite subgroup of a
projective torus. For finite fields, it is shown that X is a subgroup of a
projective torus if and only if X is parameterized by monomials. We express the
regularity of the vanishing ideal over a bipartie graph in terms of the
regularities of the vanishing ideals of the blocks of the graph.Comment: Bull. Braz. Math. Soc. (N.S.), to appea
Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras
The present text surveys some relevant situations and results where basic
Module Theory interacts with computational aspects of operator algebras. We
tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be
published in Lecture Notes in Computer Scienc
New Quantum Codes from Evaluation and Matrix-Product Codes
Stabilizer codes obtained via CSS code construction and Steane's enlargement
of subfield-subcodes and matrix-product codes coming from generalized
Reed-Muller, hyperbolic and affine variety codes are studied. Stabilizer codes
with good quantum parameters are supplied, in particular, some binary codes of
lengths 127 and 128 improve the parameters of the codes in
http://www.codetables.de. Moreover, non-binary codes are presented either with
parameters better than or equal to the quantum codes obtained from BCH codes by
La Guardia or with lengths that can not be reached by them
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