This and next week, the algebra group will host Vincent Neiger as a visitor. He previously was a postdoc in the algebra group, but now is an Associate Professor (Maître de Conférences) at Sorbonne University, Faculty of Science and Engineering. Vincent is an expert in computer algebra and has put his name to various very fast algorithms. We look forward to working with him!
Each year the Ph.D. school of DTU Compute organizes a Ph.D. bazar, where among other things Ph.D. students can present their work. Ph.D. student Lara Vicino presented her work with her supervisors on the Weierstrass semigroups of maximal curves and illustrated the use of numerical semigroups in general. Here you can see a picture of her in action today.
In the period 13th June till 3rd July, the algebra group will be visited by Prof. Sudhir Ghorpade (IIT Bombay) and Assistant Prof. Mrinmoy Datta (IIT Hyderabad). The goal is to continue the study of a famous class of algebraic geometry codes, the projective Reed-Muller codes.
Yesterday, Leonardo Landi obtained his Ph.D. degree! The defense of his Ph.D. thesis “Semigroups, curves and AG codes” started with him giving an interesting talk about his work. After that, he answered in a very convincing way all questions posed by the members of his Ph.D. committee.
We would like to congratulate Leonardo and wish him good luck with his future career!
A preprint on fast decoding of AG codes is now available on arXiv here. It was written by Peter Beelen, Johan Rosenkilde, and Grigory Solomatov. In the preprint, it is explained how to decode any AG code fast. In fact, the obtained decoding complexity is at least as fast as any previously known algorithm even when specified to particular AG codes such as the highly studied one-point Hermitian codes. It should be said however, that for Reed-Solomon codes, which can be viewed as AG codes as well, there exists a slightly faster decoding algorithm.
The result in the preprint may well become the bench mark for any future work on decoding AG codes. They also appeared as a chapter in the recent PhD thesis of Grigory Solomatov. Needless to say that we are very excited by the obtained results!
Update (November 2022): the paper has appeared in the highly esteemed journal IEEE Transactions on Information Theory!
The article “On the constant D(q) defined by Homma” by the algebra group members Peter Beelen, Maria Montanucci, and Lara Vicino has been accepted for publication in the Proceedings of the 18th Conference on Arithmetic, Geometry, Cryptography, and Coding Theory in the AMS book series Contemporary Mathematics (CONM). In this article, we investigate the asymptotic behaviour of the number of points on an algebraic curve defined over a finite field in relation to its degree. A preprint of this article can be found here.
Today, Grigory defended his Ph.D. thesis successfully! After giving an excellent presentation on his work, he answered various questions from his opponents. During his Ph.D. Grigory obtained various results on computational aspects of algebraic geometry (AG) codes, including results on encoding and decoding such codes. The results include decoding any AG code very efficiently, matching and very often improving preexisting decoders.
We would like to acknowledge the support from The Danish Council for Independent Research (DFF-FNU) for the project Correcting on a Curve, Grant No. 8021-00030B. Their support made Grigory’s Ph.D. project and the obtained results a reality.
Congratulations to Grigory with obtaining his Ph.D.-degree!
Yesterday, Grigory Solomatov handed in his PhD thesis “Computational Aspects of Algebraic Geometry Codes”. We would like to congratulate Grigory with this! His PhD defense is planned to take place on the 16th of December.
Grigory’s PhD was financed by the DFF-FNU project “Correcting on a curve”. Grigory will stick around a bit longer, since it was possible to hire him as a research assistant till the end of the year.
The paper “Point-line incidence on Grassmannians and majority logic decoding of Grassmann codes” by Peter and Prasant Singh appeared online in the journal Finite Fields and Their Applications. In this paper Peter and Prasant, a former postdoc of the algebra group, explain how to use the subtle structure of lines on Grassmannians to correct errors using Grassmann codes with an old technique called majority logic decoding. The result is a fast decoding algorithm for Grassmann codes.
Peter and Maria participated from 31st May – 4th June 2021 in the biannual conference AGCT, where researchers from all over the world discuss algebraic geometry/number theory and their applications. This year the conference was held online, due to the covid19 situation. Peter gave a talk “On a conjecture by Sørensen” on Thurday, 3rd June, in which he explained his work with Mrinmoy Datta and Masaaki Homma on the maximal number of intersection points on a non-degenerate Hermitian variety and a surface of degree d. This work has appeared recently in Proc. Amer. Math. Soc. 149 (2021), no. 4, 1431–1441. A preprint is available here.