Codes from Incidence Geometries

Vortrag von Prof. Dr. Luca Giuzzi

Sprecher eingeladen von: Prof. Dr. Joachim Rosenthal

Datum: 04.03.26  Zeit: 15.15 - 16.15  Raum: Y27H28

Incidence geometries are a powerful and general framework to axiomatically describe classical geometric objects, in terms of relations between objects of different types. Important cases of incidence geometries are induced by point -- line geometries, i.e. geometries determined by two sets \({\cal P}\neq\emptyset\) and \(\cal L\) (called respectively points and lines) and an incidence relation such that and for any two given distinct points there is at most a line which is incident with them both. In a point-line geometry, we say that two points \(p\) and \(q\) are collinear (written \(p\perp q\)) if there is a line \(\ell\in{\cal L}\) such that both \(p\) and \(q\) are incident with \(\ell\).

In this talk we shall focus mostly on \(\Gamma\)-geometries i.e. point--line geometries where given \((p,\ell)\in{\cal P}\times{\cal L}\) we have either \(\forall q\in\ell: p\perp q\) or \(|\{ q\in\ell: p\perp q\}|\leq 1\). Projective spaces, polar spaces as well as their grassmannians are all \(\Gamma\) geometries.

Next, we shall discuss the notion of projective embeddings as providing useful "concrete" models for geometries, by representing their points as varieties embedded in a suitable projective space.

By regarding embedded geometries as  projective systems it is possible to construct codes which usually admit large automorphism groups and sport a rich "local" structure which can be used in order to perform efficient decoding.

In this talk, I will report on the general constructions and then focus on the case of polar Grassmannians and of the representation of flag geometries of a given geometry.

All of these results are joint work with Ilaria Cardinali from University of Siena.

References

[1] Ilaria Cardinali and Luca Giuzzi, Codes and caps from orthogonal Grassmannians, Finite Fields and their Applications 24 (2013) 148–169.
[2] ________, Enumerative Coding for Line Polar Grassmannians with Applications to Codes, Finite Fields and their Applications 46 (2017) 107–138.
[3] ________, Linear codes arising from the point-hyperplane geometry – part I: the Segre embedding, Finite Fields and their Applications 111 (2026) 102766.
[4] ________, Linear codes arising from the point-hyperplane geometry – part II: the twisted embedding, arXiv:2507.16694 (2025).
[5] ________, On minimal codes arising from projective embeddings of point–line geometries, arXiv:2511.22747 (2025).