Title: On codes and designs

Abstract:

This is a slightly extended version of a talk I gave at the ENCODE
workshop in Eindhoven in June 2026.

We investigate connections between block codes and combinatorial
designs, including Delsarte's algebraic perspective on these objects in
the framework of association schemes and his later approach in terms of
finite regular meet-semilattices.
In this setting, the study of intersection numbers, together with an
extension of the associated Mendelsohn equations, yields a unified
treatment of the distance distributions of MDS, MRD, and other kinds of
extremal codes. It also leads to a new characterization of designs and,
derived from it, a new bound of linear programming type.

Finally, we investigate the recently discovered Steiner system with
parameters 3-(42,6,1). Its automorphism group is isomorphic to the
affine linear group AGL(2,3) and suggests a geometric description based
on the affine plane of order 3.