Parameterized codes over weighted projective spaces

In this thesis we study parametric codes defined by any subset of a weighted projective space which is parameterized by monomials. The methods and notions used to study these codes are presented in detail. This work consists of five main chapters. In the first chapter varieties and ideals in affine spaces are introduced and then their vanishing ideals that we will use in studying the codes on the varieties are explained. In this chapter numerical semigroups and Arf numerical semigroups which will be used in defining weighted projective spaces are explained. In the second chapter, graded rings, ideals and modules are explained in order to understand better the weighted projective spaces and their subvarieties. In this chapter Hilbert series that gives the value of $\alpha$-invariant that we consider when calculating parameters of codes are explained and the relation between Hilbert series and free resolutions is given. In the third chapter, weighted projective space is defined and its properties are given. In this section projective space which is an example of weighted projective space is also given. In the fourth chapter, weighted projective torus is defined and some of its properties are given. In this section we give a theorem in literature about how the vanishing ideal of a weighted projective torus is obtained and Hilbert series of this ideals is calculated. In the fifth chapter, parameterized codes in a weighted projective space that is the main aim of this thesis are explained. In this section we present how the parametric codes in a weighted projective space are constructed and share some examples. We prove an observation hinted by these examples. In the examples section the parameters of parametric codes which are parameterized by different subsets in weighted projective spaces determined by different numerical semigroups are given and whether the codes are good is tried to be understood.