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.
The article “On algebraic curves with many automorphisms in characteristic p” by the algebra group member Maria Montanucci has been accepted for publication in Mathematische Zeitschrift.
In this article, Maria gives a partial answer to an open problem regarding the size and the action of large automorphism groups of agebraic curves in positive characteristic. A preprint of this article can be found here
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.
A preprint on generalized Weierstrass semigroups on a family of maximal curves is now available on arXiv here.
It was written by algebra group member Maria Montanucci and Guilherme Tizziotti from Universidade Federal de Uberlândia (UFU), Brazil. In the preprint the generalized Weiestrass semigroup at several points of a family of maximal curves which is not covered by the Hermitian curve is computed.
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!
The paper “A class of linear sets in PG(1,q^5)” by Maria and Corrado Zanella appeared online in the journal Finite Fields and its Applications. In the manuscript Maria and Corrado study some interesting combinatorial structures called maximum scattered linear sets over projective spaces. These structures have been intensively studied during the last year, particularly for the connection to coding theory (MRD codes).
Maximum scattered linear sets over PG(1,q^n) have been completely classified for n at most 4 by Csajbók-Zanella (2018) and Lavrauw-Van de Voorde (2010). In this paper Maria and Corrado analyze the case n=5. There a wide class of linear sets is studied which depends on two parameters. Conditions for the existence, in this class, of possible new maximum scattered linear sets in are exhibited.
The paper “An Fp2-maximal Wiman’s sextic and its automorphisms” by Maria, Massimo Giulietti, Motoko Kawakita and Stefano Lia appeared online in the journal Advances in Geometry. In this paper Maria, Massimo, Motoko and Stefano study a sextic curve introduced in 1895 by Wiman as a Riemann surface over the complex field, but seen over finite fields. They showed that its full automorphism group is isomorphic to the symmetric group S5, generalizing Wiman’s result. It is also shown that when the finite field has cardinality 19^2 then the Wiman’s sextic is maximal and it is not Galois covered by the Hermitian curve.
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.