Singularities as generators of signal constellations and hyperbolic triangles

Abstract

In this work, three regular singularities of fuchsian differential equations will be considered, one of them being infinity, as generators of signal constellations and as vertices of a hyperbolic polygon, in order to establish a connection with error analysis in a discrete memoryless channel and in a geometrically uniform code. The procedure is based on the following steps: given two complex points and the infinite: (1) consider these points as singularities of a fuchsian differential equation; (2) generate constellations of signals on the complex plane from the obtained singularities; (3) consider the existence of a perfect or quasi-perfect code on the signals constellation and their ability to correct errors; (4) analyze these singularities as vertices of a hyperbolic triangle to identify the genus of the associated surface, through the pairings of the sides of this triangle; (5) verify which channel is associated with the same genus, and thus represent the vertices of the hyperbolic triangle as channel inputs and outputs; (6) analyze the channel associated with the codewords to verify the error probability of the transmitted singularity. Through the results obtained, it can be concluded that singularities with the opposite real part generate perfect and/ or quasi-perfect codes with the same ability to correct errors. It was possible to represent the codewords as inputs and outputs of a discrete memoryless channel, showing that the probability of error, p, is related to the number of codewords on the constellation. When considering singularities as vertices of a hyperbolic triangle, a connection was established with the symmetric binary channel C_2.2, whose inputs and outputs represent pairs of opposite singularities relative to the imaginary axis. The established connections can be applied to error analysis in an information transmission process.