Conexões existentes entre equações diferenciais fuchsianas, geometria hiperbólica e códigos corretores de erros, aplicadas em canais discretos sem memória

Abstract

The Theory of Error-Correcting Codes, Hyperbolic Geometry and Fuchsian Differential Equations have been becoming present in the work of several researchers, and they are areas with several possible applications, such as in the study of errors that may occur in the process of information transmission. In this work, we present possible connections between geometrically uniform codes, hyperbolic geometry elements and Fuchsian differential equations, obtained through theoretical studies with the main definitions and properties of the respective areas. Complex singularities of Fuchsian differential equations that also generated signal constellations in the complex plane were considered, then the existence of a perfect or quasi-perfect code was analyzed, which presented the same error correction capability, regardless of the singularity generating the code. Finally, it was possible to represent the codewords as inputs and outputs of a discrete channel without memory, showing that the error probability, p, is related to the number of codewords on the constellation. Another path established was to analyze these singularities as vertices of a hyperbolic triangle to analyze the genus of the associated surface, through the pairings of the sides of this triangle, besides establishing a connection with the symmetric binary channel C2,2, verifying that the error probability is also the same, regardless of the transmitted singularity. In addition, we present new perfect and quasi-perfect codes on quotient rings of Gaussian integers, highlighting a different geometric structure from that found in the literature. The results exposed in this work contribute to the development of tools that can be applied both in mathematics and in engineering, since algebraic and information theory concepts were utilized, areas in booming expansion.