ACCESS (Algebraic Coding and Cryptography on the East Coast) is a recently launched online seminar series designed to highlight world-class research in coding theory, cryptography and related areas and to encourage collaboration between its participants, see here. As invited speaker, Maria gave a talk today with the title “Maximal curves over finite fields and their applications to coding theory”. The slides of the presentation are available here.
As part of the DTU outreach activities towards high schools, Peter gave a guest lecture on solving cubic polynomial equations. The activities are organized within the framework of an InterMat workshop (see https://matematicum.dtu.dk/intermat for information. Danish only). With various high school students listening online, an explanation was given on how to use Tschirnhaus transforms to obtain the famous Cardano solution formulas. The same method can be used to solve the quartic equation.
Today Maria gave a seminar for the online seminar series “Galois geometries and their application”, organized by Olga Polverino, Ferdinando Zullo and Giovanni Zini (University of Campania, Caserta). The title of her seminar was “Maximal curves over finite fields” and the slide are available here. The webpage of the conference can be found at this link.
Today Maria’s joint work with Daniele Bartoli and Luciane Quoos “Locally recoverable codes from automorphism groups of function fields of genus g>=1” appeared in the journal IEEE Transactions on Information Theory. Here locally recoverable codes from function fields of genus at least 1 are constructed and a new Singleton-like bound for locally recoverable codes with multiple recovering sets is obtained. A preprint can be found here.
On the 15th of October, a new PhD student named Lara Vicino started in the algebra group. She will be supervised by Peter Beelen and co-supervised by Maria Montanucci. Her project is about determining upper bounds on the number of points on a curve defined over a finite field in terms of the degree of the curve.
This month, the paper “Hyperplane Sections of Determinantal Varieties over Finite Fields and Linear Codes” appeared in the September issue of the journal Discrete Mathematics. The paper is written by Peter and Prof. Sudhir Ghorpade from IIT Bombay, a regular guest of the algebra group with whom Peter has written several papers before. A preprint can be found here.
The paper “Equivalence and Characterizations of Linear Rank-Metric Codes Based on Invariants” [arXiv] by Alessandro Neri (TU Munich, Germany), group member Sven Puchinger, and Anna-Lena Horlemann-Trautmann (University of St. Gallen, Switzerland) was accepted for publication in Linear Algebra and Its Applications.
The paper studies a tool to efficiently determine the equivalence of linear rank-metric codes – a topic of growing importance due to many new code constructions in the last years. Furthermore, bounds on the number of equivalence classes of two code families and different characterizations for one code class are given.
Today Maria and Peter published their paper “On subfields of the second generalization of the GK maximal function field” in Finite Fields and their Applications. Here several new maximal curves are constructed corresponding to subfields of the second generalized GK maximal curve. A preprint can be found here.
On May 28-29, Peter co-organized a workshop with Fabien Pazuki from Copenhagen University. The workshop was part of a series of activities organized by N^3 (Nordic Number theory Network, see here for more information). Though originally planned to take place at DTU, the event was held online due to the covid19 situation.
Maria and Giovanni Zini published their joint paper “The complete list of genera of quotients of the Fq2-maximal Hermitian curve for q≡1 (mod 4)” in Journal of Algebra. Here an open problem from 2000 is partially solved, namely that of computing all the genera of quotients of the Hermitian curve over Fq2, q prime power. Indeed such a complete list of genera is obtained here if q is congruent to 1 modulo 4. A preprint can be found here.