Public defence in Information Theory, M.Sc. Mikhail Ganzhinov

The title of the thesis: Construction of few-angular spherical codes and line systems in Euclidean spaces

Thesis defender: Mikhail Ganzhinov
Opponent: Prof. Peter Boyvalenkov, Bulgarian Academy of Sciences, Bulgaria
Custos: Prof. Patric Östergård, Aalto University School of Electrical Engineering

This thesis investigates the construction of few-angular spherical codes and line systems using various algebraic and combinatorial methods. The most important algebraic method is automorphism prescription, while combinatorial methods include the exhaustive isomorph-free generation of Gram matrices representing spherical codes and weighted clique search in graphs with vertices representing orbits of vectors. 

A spherical code is an arrangement of a finite number of unit vectors in n-dimensional Euclidean space. In an antipodal spherical code, for each vector, its opposite vector is also included in the set. Pairs of opposite vectors define lines, hence antipodal spherical codes are also called line systems. 

Spherical codes have numerous applications in communications, where they are used to optimize error correction and spectral efficiency of signals. Applications typically require codes with well-spaced vectors. For example, in an additive white Gaussian noise (AWGN) channel with high enough signal to noise ratio, spherical codes with the largest angular distance between nearest vectors minimize error probability. Such codes are called optimal. A spherical code or a line system is called few-angular if the number of distinct angular distances between vectors or lines of the code is small. Many of the known optimal and nearly optimal spherical codes and line systems are few-angular. 

The obtained results include the classification of the largest systems of real biangular lines for dimension d≤6 and the construction of two infinite families of line systems that achieve equality in the second Levenshtein bound. Several low-dimensional spherical codes with large minimum angular distances between the vectors were constructed, with three of these codes improving the lower bounds for kissing numbers in dimensions d=10,11 and 14.

Thesis available for public display 7 days prior to the defence at Aaltodoc.