Projective geometry | EPFL Graph Search

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an e

Examples of normal, not geometrically normal, projective Gorenstein del Pezzo surfaces with at most Du Val singularities

Ursina Schweizer

While over fields of characteristic at least 5, a normal, projective and Gorenstein del Pezzo surface is geometrically normal, this does not hold for characteristic 2 and 3. There is no characterization of all such non-geometrically normal surfaces, but there is a complete characterization of all possible surfaces that can arise as the normalization of the base change to the algebraic closure of the base field. In four of these instances, for a given normalization, we describe the construction of such a non-geometrically normal surface.

EPFL2021

Algebraic methods for channel coding

Frédérique Oggier

This work is dedicated to developing algebraic methods for channel coding. Its goal is to show that in different contexts, namely single-antenna Rayleigh fading channels, coherent and non-coherent MIMO channels, algebraic techniques can provide useful tools for building efficient coding schemes. Rotated lattice signal constellations have been proposed as an alternative for transmission over the single-antenna Rayleigh fading channel. It has been shown that the performance of such modulation schemes essentially depends on two design parameters: the modulation diversity and the minimum product distance. Algebraic lattices, i.e., lattices constructed by the canonical embedding of an algebraic number field, or more precisely ideal lattices, provide an efficient tool for designing such codes, since the design criteria are related to properties of the underlying number field: the maximal diversity is guaranteed when using totally real number fields and the minimum product distance is optimized by considering fields with small discriminant. Furthermore, both shaping and labelling constraints are taken care of by constructing Zn-lattices. We present here the construction of several families of such n-dimensional lattices for any n, and compute their performance. We then give an upper bound on their minimal product distance, and show that with respect to this bound, existing lattice codes are optimal in the sense that no further significant coding gain could be reached. Cyclic division algebras have been introduced recently in the context of coherent Space-Time coding. These are non-commutative algebras which naturally yield families of invertible matrices, or in other words, linear codes that fulfill the rank criterion. In this work, we further exploit the algebraic structures of cyclic algebras to build Space-Time Block codes (STBCs) that satisfy the following properties: they have full rate, full diversity, non-vanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We give algebraic constructions of such STBCs for 2, 3, 4 and 6 antennas and show that these are the only cases where they exist. We finally consider the problem of designing Space-Time codes in the noncoherent case. The goal is to construct maximal diversity Space-Time codewords, subject to a fixed constellation constraint. Using an interpretation of the noncoherent coding problem in terms of packing subspaces according to a given metric, we consider the construction of non-intersecting subspaces on finite alphabets. Techniques used here mainly derive from finite projective geometry.

EPFL2005

Algebraic methods for channel coding

Frédérique Oggier

EPFL2005