Spherical codes with cyclic symmetries

Author(s)

Rogério Monteiro de Siqueira

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2006-05-18

Examining board members

Sueli Irene Rodrigues Costa; Maria Aparecida Soares Ruas; Cecilio Jose Lins Pimentel; Marcelo Muniz Silva Alves; Reginaldo Palazzo Júnior

• Advisor: Sueli Irene Rodrigues Costa

Abstract

Euclidean spherical codes with symmetries are orbits of finite orthogonal matrix groups. These codes are also known as group codes. ln this work, the commutative group codes in even dimensions are viewed on flat tori, which are submanifolds of the sphere. Also, if the matrix group is cyclic, the generated code lies on a knot which wraps around a torus. If the dimension is odd, every commutative group code lies on an anti-prism whose bases are contained in two flat tori. This interpretation lead us to build upper bounds for the cardinality of these constellations involving their minimum distance and the packing density of an associated lattice. Using a method by Biglieri and Elia, which searchs the initial vector for a cyclic group in order to achieve the best minimum distance, we also present the best cyclic group codes in dimension four up to 100 points (AU)