Projective codes meeting the Griesmer bound

AbstractWe present a brief survey of projective codes meeting the Griesmer bound. Methods for constructing large families of codes as well as sporadic codes meeting the bound are given. Current research on the classification of codes meeting the Griesmer bound is also presented

Characterization results on weighted minihypers and on linear codes meeting the G(r)iesmer bound

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Linear codes meeting the Griesmer bound, minihypers and geometric applications

Coding theory and Galois geometries are two research areas which greatly influence each other. In this talk, we focus on the link between linear codes meeting the Griesmer bound and minihypers in finite projective spaces. Minihypers are particular (multiple) blocking sets. We present characterization results on minihypers, leading to equivalent characterization results on linear codes meeting the Griesmer bound. Next to being interesting from a coding-theoretical point of view, minihypers also are interesting for geometrical applications. We present results on maximal partial μ-spreads in PG(N, q), (μ + 1)|(N + 1), on minimal μ-covers in PG(N, q), (μ + 1)|(N + 1), on (N − 1)-covers of Q + (2N + 1, q), on partial ovoids and on partial spreads of finite classical polar spaces, and on partial ovoids of generalized hexagons, following from results on minihypers

The use of blocking sets in Galois geometries and in related research areas

Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems

Applications of finite geometry in coding theory and cryptography

We present in this article the basic properties of projective geometry, coding theory, and cryptography, and show how finite geometry can contribute to coding theory and cryptography. In this way, we show links between three research areas, and in particular, show that finite geometry is not only interesting from a pure mathematical point of view, but also of interest for applications. We concentrate on introducing the basic concepts of these three research areas and give standard references for all these three research areas. We also mention particular results involving ideas from finite geometry, and particular results in cryptography involving ideas from coding theory