## Simitli Çeşitlem Üzerinde Üzerinde Parametrik Kodlar Ve Sıfırlayan İdealler

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2021##### Author

Baran Özkan, Esma

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Let X be a complete simplicial toric variety over a finite field with a split torus T_X.
This thesis is on parameterized codes obtained from the subgroups of the torus T_X
parameterized by matrices. It is very important to find the generators of the vanishing
ideals of these subgroups to compute basic parameters of these codes. In the introduction
part of the thesis, the significance and a literature review of toric codes are given.
The results obtained in the thesis are summarized. The second chapter includes some
background of affine varieties required for the affine toric varieties.
In the third chapter, after defining torus, the concepts of character and one parameter
subgroup of a torus are presented, and are associated with lattices. After giving the
definition of toric variety, the different constructions of affine toric varieties are explained.
The basic topics of rational polyhedral cones are given and, their connections its
with affine toric varieties is explained.
In the fourth chapter, by gluing affine varieties with isomorphisms, abstract varieties
other than affine or projective varieties are constructed. This chapter starts with
projective varieties for a better understanding of these varieties. How to glue affine toric
varieties corresponding to elements in a finite collection of strong rational polyhedral
cones, called fan, is described, and so general toric varieties are constructed. The main
and final purpose of this section is to show that the points of a general toric variety
can be expressed with homogeneous coordinates as in projective space.
For a given matrix Q, denote by T_{X,Q} the subgroup of the torus T_X parameterized by
the columns of Q. In the fifth chapter, 3 algorithms are given to determine a generating
set of the vanishing ideal of T_{X,Q}. Elimination theory is used in the first algorithm
developed. A Macaulay2 code is written to implement the algorithm. Another method
for finding the generators of the same ideal using the base of the lattice describing this
ideal is obtained. An algorithm for finding the lattice L such that I(T_{X,Q}) = I_L and a
procedure implementing this algorithm in the Macaulay2 program is presented. Thus,
it is easily checked whether the vanishing ideal I(T_{X,Q}) is a complete intersection or not.
In this section, finally, a method for conceptually determining the lattice L is obtained
and a Nullstellensatz Theorem is proven on a finite field under some conditions.
The sixth chapter constitutes the heart of the thesis and includes parameterized
codes constructed by calculating homogeneous polynomial functions in the set T_{X,Q}.
For this purpose, firstly, basic topics of linear codes are explained. Since the dimension
of parametric code C_{\alpha_Q} is calculated with multigraded Hilbert function of toric set
T_{X,Q}, some properties of multigraded Hilbert functions are given. Using parametric
definition of the T_{X,Q}, an algorithm directly computing the number of elements of the
subgroup T_{X,Q} which is equal to the length of the code and a lower bound for the
minimum distance of the code is obtained. As an application, the basic parameters of
the parameterized codes obtained from the torus of the Hirzebruch surface are calculated.
Finally, examples illustrating the advantage of passing from projective space to
arbitrary toric variety, in addition to working with parameterized toric set T_{X,Q} instead
of the torus T_X are given.

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