Today Maria’s paper “On a conjecture about maximum scattered subspaces of Fq6×Fq6” appeared online in Linear Algebra and its Applications, as a joint work with Daniele Bartoli and Bence Csajbók. Here Bence, Daniele and Maria proved a conjecture from 2018 regarding a family of maximum scattered subspaces of Fq6 x Fq6. A preprint is available here.
Today, Maria gave a talk for the online conference SIAM Conference on Applied Algebraic Geometry (AG21). Her talk, titled “Error correcting codes from maximal curves”, was a part of the mini-symposium Polynomial and Evaluation Codes and their Applications. In her talk Maria presented several construction of error correcting codes (AG codes, quantum codes and locally recoverable codes) using the nice structure of automorphism groups of maximal curves.
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.
Today, Maria’s paper “Fp2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve” appeared in the journal Advances in Geometry, as a joint work with Daniele Bartoli and Fernando Torres. Here a open problem dated 2000 is analyzed. Namely, is it true that every Fp2-maximal curve (where p is a prime) is covered by the Hermitian curve? In this paper Maria, Daniele and Fernando show that this is the case provided that the curve has a sufficiently large automorphism group. Also, the first example of an Fp2-maximal curve which is not Galois-covered by the Hermitian curve is presented. A preprint is available here.
Peter, Maria and Leonardo’s paper “Weierstrass semigroups on the Skabelund maximal curve” has been accepted for publication in the journal Finite Fields and their Applications. Here the Weierstrass semigroup is computed at every point of the Skabelund maximal curve (covering the famous Suzuki curve), as well as the Weierstrass points of the curve. Since these are the main ingredients to construct AG codes, the result has interesting consequence in applications. Also, this is one of the few examples in the literature in which this complete analysis is known. A preprint is available here.
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.
Peter and Maria participated in the online conference “Curves over Finite Fields”, which took place 24-28 May. The conference was originally planned to take place in Benasque, but was held online, due to the covid19 situation.
Peter was an invited plenary speaker and gave two talks: “Asymptotic results: an overview” and “Asymptotic results: what is next?”. In the first talk, he gave an overview of results on asymptotically families of curves over a finite field with many rational points. In the second talk, several open problems and very recent developments were explained.
Maria was an invited semi-plenary speaker and gave the talk “On the automorphism group of algebraic curves in positive characteristic”. In the talk, she gave an overview on the latest developments on relation between various invariants of algebraic curves, such as p-rank, genus, and their automorphism groups.
We are happy to say that the given talks were well received!
Today Maria’s paper “On certain self-orthogonal AG codes with applications to Quantum error-correcting codes” appeared in the journal Designs, Codes and Cryptography, as a joint work with Daniele Bartoli and Giovanni Zini. Here, a construction of quantum codes from self-orthogonal algebraic geometry codes is provided as well as on a new family of algebraic curves to which the construction is applied, named Swiss curves. The reason behind this curious name, is that the authors developed the key ideas proposed in this publication during the conference SIAM AG19, which took place in Bern (Switzerland). A preprint is available here.
Peter has been one of the lectures in an international workshop organized by leading technical universities in India: IIT Bombay, IIT Jammu and SGGS-IET. The online event took place March 8-12 and had many participants from all over the world. Peter gave two lectures on the decoding of Reed-Solomon codes. Slides of his and other talks are available here.
In the issue of this month of the prestigious journal IEEE Transactions on Information Theory, you can find the article “Fast Encoding of AG Codes over C_ab Curves” by Peter, Johan, and Grigory. In this article it is explained how to convert a message into a codeword (encoding) as well as the reverse process (unencoding) fast for a significant class of algebraic geometry codes. A preprint is available here.