Vanishing ideals and parameterized codes on toric variety

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.