Peter Beelen and Maria Montanucci from the algebra group have visited the Antalya Algebra Days, which this year took place in the Nesin mathematical village in Turkey from the 15th till the 19th of May. Peter gave a talk about his recent research on generalized Hamming weights of Reed-Muller like codes, some of which the carried out together with former postdoc Mrinmoy Datta.
On the 1st of April 2019, Maria Montanucci joined the algebra group in the section for mathematics at DTU Compute as assistant professor. We all wish her welcome!
Maria studied mathematics at the Universitá degli Studi di Perugia in Italy, obtaining her bachelor’s degree cum laude in 2013 and her master’s degree cum laude in 2015. From 2015-2018, she did her PhD under the supervision of Prof. G. Korchmáros at the Università della Basilicata, Italy. After this she was a postdoc at the University of Padua from November 2018 – February 2019, where she worked with Prof. Corrado Zanella.
Today, Peter gave the inaugural lecture for a conference celebrating the influence of Ruud Pellikaan in mathematics on the occasion of his retirement. Ruud Pellikaan was Peter’s PhD supervisor and it was a great pleasure for him to give an overview of Pellikaan’s contributions to mathematics. We wish Ruud a happy retirement!
Prof. Trygve Johnsen from The Arctic University of Norway will visit the algebra group in the section for mathematics at DTU Compute in the coming three months. He is currently on a sabbatical and fortunately for us, he chose DTU as one of the universities he would like to visit. During his stay, he will primarily work with Peter Beelen and Prasant Singh on Grassmann and Schubert codes.
From October 27 til November 1, Peter Beelen visited Prof. Massimo Giulietti and his team at the Università degli Studi di Perugia in Italy. During the visit he finished an article with Maria Montanucci, who visited DTU as a guest PhD in 2017. Directly after the visit, Maria started as a postdoctoral researcher at the University of Padua.
In the joint article certain subfields are described of the new maximal function fields found by Maria and Peter in 2017. Such subfields are of interest, since by a result of Serre, they are maximal as well. The article has been submitted to the journal Finite Fields and Their Applications. A preprint is available here.
On October 15, PhD student Grigory Solomatov joined our team! Grigory did a dual master at NTNU and DTU. In the coming three years, he will work on the decoding of AG-codes. His project is an important part of the FNU-project “Correcting on a Curve”.
Danmarks Frie Forskningsfond (Independent Research Fund Denmark) each year supports many prestigious research projects. They have just announced that they will support a research project of the algebra group. This grant will enable us to hire a Ph.D. student as well as to visit and invite researchers from all over the world. We are very grateful for the support!
A brief summary of the research project is the following:
Error-correcting codes ensure reliability in e.g. satellite communication, distributed storage and cloud computing. It is vital that the error-correction can be done fast. Excellent error-correcting codes can be constructed using algebraic curves – AG codes – but no fast decoding algorithms are available for these codes. This has limited their theoretical impact and prohibited any practical impact. This project aims to change that.
Decoding of AG codes boils down to solving a Padé approximation on the algebraic curve. Current techniques throw away many levels of structure, leaving a problem on univariate polynomials. We will seek algorithms that retain these levels, and we will build on the latest developments in computer algebra. We expect improvements for both the simplest as well as more complex AG codes.
The project will be carried out by the algebra group at DTU and international collaborators from France and Canada.
From April 15 til May 8, Prof. Masaaki Homma from Kanagawa University visits the algebra group. He will work with Peter Beelen and Mrinmoy Datta on problems related to the intersection of a Hermitian surface and a surface of degree d. The aim is to solve a conjecture posed by a Danish Ph.D. student A.B. Sørensen in 1991. For d=1 and d=2 the conjecture is known to be true, while Peter and Mrinmoy recently made significant progress for d=3, see this previous post. For d>3 nothing is known as yet, but this may change soon!
As announced in a previous post, Peter Beelen and guest Ph.D. Maria Montanucci recently solved a conjecture on the structure of Weierstrass semigroups of points on the Giulietti-Korchmaros maximal curve. These results have now appeared in the journal Finite Fields and Their Applications. Click here to see the article.
Peter Beelen and Mrinmoy Datta have just submitted an article in which a conjecture from 1991 by Sørensen is partially resolved. In this conjecture a formula for the minimum distance of q^2-ary codes constructed from Hermitian surfaces is given. Equivalently, the conjecture gives a formula for the maximal number of GF(q^2)-rational points that a Hermitian surface and a hypersurface of degree d>1 can have. For d=2, the conjecture was solved in 2007 by Edoukou, but since then no progress has been made until now. We show that the conjecture is correct for d=3 as long as q>7. A preprint can be found here.