Linear Sperner families and strong blocking sets in coding theory
- Alessandro Neri (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany)
Minimal linear codes were first introduced by Cohen and Lempel over the binary field under the name of linear intersecting codes. They later gained interest due to their application to secret sharing schemes proposed by Massey. Concretely, minimal linear codes are subspaces of vectors over a finite field such that the set of nonempty supports forms an antichain with respect to the set inclusion. In other words, they provide the vector space analogue of Sperner families. Recently, it has been shown that minimal linear codes in are in one-to-one correspondence with strong blocking sets, which are special sets of points in a projective space, such that their intersection with each hyperplane generates the hyperplane itself.
In this talk we will revise the basic theory connecting these three objects with combinatorial, algebraic and geometric flavours: linear Sperner families, minimal linear codes and strong blocking sets. We will show first properties and bounds on their cardinality and parameters. Later, we will go through some classical constructions mainly obtained as union of lines. If time allows, we will then explore new ideas for constructing strong blocking sets over any field, obtained as union of rational normal curves.