Galois geometries and applications

Galois geometries and coding theory

Galois geometries and coding theory are two research areas which have been interacting with each other for many decades. From the early examples linking linear MDS codes with arcs in finite projective spaces, linear codes meeting the Griesmer bound with minihypers, covering radius with saturating sets, links have evolved to functional codes, generalized projective Reed–Muller codes, and even fu...

Intertwined Results on Linear Codes and Galois Geometries

Programming from Galois Connections — Principles and Applications

Problem statements often resort to superlatives such as in eg. “. . . the smallest such number”, “. . . the best approximation”, “. . . the longest such list” which lead to specifications made of two parts: one defining a broad class of solutions (the easy part) and the other requesting one particular such solution optimal in some sense (the hard part). This report introduces a binary relationa...

Symmetries, Differential Equations, and Applications: Galois Bicentenary

This special issue commemorated the bicentenary of the birth of Évariste Galois who introduced the notion of a group and made fundamental contributions to the solution of polynomial equations.The formal mathematical concept of symmetry thus came fromGalois theory. Symmetry was inaugurated andutilized in the solution of differential equations by Sophus Lie in the 1870s. The algebraic approach ha...

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 ...