This work develops a geometric framework for constructing quantum error-correcting codes from weighted projective and orbifold structures, integrating algebraic geometry, divisor theory, and the CSS stabilizer formalism. Beginning with weighted projective spaces and their associated height and defect structures, the study builds classical AG-codes via evaluation on divisors adapted to orbifold singularities. These classical codes are lifted to quantum codes using self-orthogonality conditions and homological constructions, yielding a class of Quantum Weighted Algebraic Geometric (QWAG) codes. A central contribution is the formulation of a refined Singleton-type bound motivated by orbifold defect terms and effective genus corrections. While the classical quantum Singleton bound is recovered in the smooth case, the orbifold setting suggests additional geometric contributions that may adjust the theoretical distance bound. The refined bound is presented with partial justification under specific geometric hypotheses and framed as a conjectural extension in full generality. The monograph further provides explicit constructions, computational implementations in Sage/Python, and illustrative examples demonstrating how weighted geometry influences code parameters. This work establishes a structured bridge between orbifold geometry and quantum coding theory, outlining both concrete constructions and open problems for further mathematical development.

翻译：本研究发展了一种几何框架，用于从加权射影空间和轨形结构构造量子纠错码，整合了代数几何、除子理论和CSS稳定子形式体系。研究从加权射影空间及其关联的高度与缺陷结构出发，通过在适应轨形奇点的除子上进行求值，构建了经典代数几何码。这些经典码通过自正交条件与同调构造被提升为量子码，从而产生了一类量子加权代数几何码。一个核心贡献是提出了一个由轨形缺陷项和有效亏格修正所启发的精化Singleton型界。虽然在光滑情形下恢复了经典的量子Singleton界，但轨形背景暗示了可能调整理论距离界的额外几何贡献。该精化界在特定几何假设下给出了部分证明，并被表述为一个完全一般性的猜想性推广。本专著进一步提供了显式构造、Sage/Python中的计算实现以及示例，展示了加权几何如何影响码参数。这项工作在轨形几何与量子编码理论之间建立了一座结构化的桥梁，概述了具体的构造方法和有待进一步数学发展的开放问题。