Holographic codes on Bruhat-Tits buildings and Drinfeld symmetric spaces (Q2295837)

The author provides the reader with an introductory perspective regarding the construction of holographic classical and quantum codes on Bruhat-Tits trees. The work is a part of a bigger project on the already mentioned topic. The paper is structured in five sections. The introduction is a very well documented starting point for the article with respect to ``the question of constructing holographic codes on \(p\)-adic symmetric spaces, based on algebro-geometric properties''. The recalled references and concepts represent the main ideas on which the article is based. In Section 2, entitled \textit{Algebro-Geometric Codes on the Bruhat-Tits tree}, Reed-Solomon codes are recalled for the author to be able to expose its theory on how to obtain holographic codes based on the Burhat-Tits tree. Classical algebro-geometric codes for Mumford curves are described as well as quantum algebro-geometric codes. Section 3, \textit{Discrete and Continuous Bulk Spaces: Bruhat-Tits Buildings and Drinfeld Symmetric Spaces}, discusses the geometry of the Drinfeld plane and higher rank buildings and Drinfeld symmetric spaces. In Section 4, entitled \textit{Tensor Networks on the Drinfeld Plane}, the pentagon code on the real hyperbolic plane is described and, then, triangle Fuchsian groups and holographic codes are tackled. Also, surface quantum codes and triangle groups on the Bruhat-Tits trees are discussed. Then, tessellations of the Drinfeld plane and the lifting of holographic codes from the Bruhat-Tits tree are presented. Section 5, \textit{Holographic Codes on Higher Rank Bruhat-Tits Buildings}, codes on the Bruhat-Tits buildings of GL\(_3\) from algebro-geometric codes on surfaces are discussed, as well as codes on Drinfeld symmetric spaces. The conclusions of the work are also presented at the end of Section 5, and the authors point out new meaningful research directions. Important references are given at the end of the paper. The technical concepts and properties introduced throughout the article are described in a clear and formal manner. The results of this work are suitable for specialized graduate students which already have technical background regarding the topic of algebro-geometric codes and, especially, researchers. We could not identify aspects which are not in accordance with the scope of the paper or other incorrect details, other than the fact that the subsections of section 3 are not numbered (as opposed to the case of all the other sections of the article). Only a few unimportant typos were spotted throughout the document. As a conclusion, we stress that the current article is a pertinent work mainly for researchers interested in technical aspects and new results regarding holographic codes, especially built upon the Bruhat-Tits tree.