Ağırlıklı Projektif Uzaylar Üzerindeki Kodlar ve Onların Cebirsel Değişmezleri

Abstract

Weighted projective spaces, when considered in light of the geometric definition for projective spaces and allowing non-trivial weights, exhibit unique structures both geometrically and algebraically. By non-trivial weights, we mean scenarios where not all the weights are equal to 1. When all the weights are 1, the structure corresponds to the classical projective space; thus, weighted projective spaces can be viewed as natural generalizations of classical projective spaces. These spaces are recognized as suitable environments for constructing interesting classes of linear codes over finite fields. Such codes are known as Weighted Projective Reed-Muller Codes. Classical Projective Reed-Muller (PRM) codes extend Reed-Muller (RM) codes, which play a crucial role in reliably transmitting information over digital communication channels. Classical PRM codes have been thoroughly studied and are used in various real-world applications. This thesis focuses on the study of Weighted Projective Reed-Muller Codes which are an extension of PRM codes. Specifically, codes over the family of weighted projective spaces of the form P(1,a,b), where a and b are positive coprime integers, have been examined, and their parameters have been calculated. First, considering the case a=1, let F_q be a finite field with q elements, Y=P(1,1,b)(F_q) the set of F_q-rational points of the weighted projective space X=P(1,1,b). The basic parameters of the code C_{d,Y} corresponding to this set have been provided. Results concerning the dimension of the code, one of these basic parameters, have been derived through the Hilbert function which is one of the algebraic invariants. Consequently, the free resolution of the vanishing ideal I(Y) corresponding to this space, its Hilbert series, and the values of its Hilbert function have been obtained. The regularity index and hence the regularity set, which are essential for eliminating trivial codes, have also been determined. Subsequently, in addition to these results and methods, the basic parameters of the codes obtained from the space X=P(1,a,b) have been calculated, considering both geometric and combinatorial methods. Formulas for calculating the code's dimension have been provided, referencing the relationship between the lattice points of the corresponding polygon and the dimension of the code. Additionally, using a bound known in the literature as the footprint bound, and also based on Gröbner basis theory, a lower bound for the minimum distance of the code obtained from this space has been provided. The regularity set and the regularity index of the set of rational points corresponding to this space have also been determined. Therefore, the main parameters of the codes obtained from the weighted projective space X=P(1,a,b) and other related geometric and algebraic results have been obtained within the scope of this thesis.