DOCUMENTA MATHEMATICA, Extra Volume ICM III (1998), 397-408

Joseph A. Thas

Title: Finite Geometries, Varieties and Codes

In recent years there has been an increasing interest in finite projective spaces, and important applications to practical topics such as coding theory and design of experiments have made the field even more attractive. It is my intention to mention some important and elegant theorems, to say something about the used techniques and the relation with other fields, and to mention some open problems. First some characterizations of particular pointsets in the projective space PG$(n,q), n \geq 2$, over GF$(q)$ will be given, where, from the beginning, it is assumed that the pointset is contained in PG$(n,q)$. A second approach is that where the object is described as an incidence structure satisfying certain properties; here the geometry is not a priori embedded in a projective space. This approach will be illustrated with some theorems on inversive planes, polar spaces and Shult spaces. Finally, there is a section on $k$-arcs in PG$(n,q)$ and on linear Maximum Distance Separable codes, where the interplay between finite projective geometry, coding theory and algebraic geometry is particularly present. In an appendix an example of brand new research in the field is given.

1991 Mathematics Subject Classification: 51E15, 51E20, 51E21, 51E25, 51A50, 51B10, 05B05, 05B25, 94B27

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