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.
The article Counting Generalized Reed-Solomon Codes by Peter Beelen, David Glynn, Tom Høholdt and Krishna Kaipa has just appeared in the journal Advances in Mathematics of Communication, volume 11, issue 4, pages 777-790, doi: 10.3934/amc.2017057. The work started by a question of the fourth author posed at a conference in India in 2013 and grew out to a nice journal article in the years after. You can see an updated preprint of the article here.
In a new preprint Peter Beelen and guest Ph.D. Maria Montanucci describe a new family of maximal curves. Maximal curves are algebraic curves defined over a finite field, having as many rational points as allowed by the famous Hasse-Weil bound. Such curves are of great interest by themselves, but are also the most obvious curves to choose when constructing algebraic-geometry (AG) codes.
Note that the depicted curves don’t have anything to do with the new maximal curves. The new curves are over a finite field and are therefore difficult to meaningfully depict graphically.
Peter Beelen and guest Ph.D. Maria Montanucci have solved a conjecture on the structure of Weierstrass semigroups for certain points on the Giulietti-Korchmaros maximal curve. This conjecture was posed in 2011 in the article Two-Point Coordinate Rings for GK-Curves, IEEE Trans. Inf. Theory 57(2), 593-600 by I. Duursma. Apart from proving the conjecture, Peter and Maria in fact determined the structure of the Weierstrass semigroup of any point on the GK curve. A preprint of these results is available here.
Maria Montanucci a Ph.D. student in Mathematics at Università degli Studi della Basilicata (Potenza, Italy) who visited the algebra group 1 April – 31 May this year, just started the second part of her stay with the algebra group at DTU. This time she will be here from 1 August – 30 November. During her last stay she obtained interesting results on maximal curves and she will continue to work on this topic with members of the algebra group.
Nurdagül Anbar, who previously has been a HCØrsted postdoc at DTU, will visit the algebra group in August. She will work on the construction of lattices from function fields. Lattices are for example used in lattice-based cryptography.
Associate professor Alp Bassa from Bogazici University, Istanbul, will visit the algebra group this week. He will work with Peter Beelen on Drinfeld modular polynomials, which have applications in the theory of asymptotically good, recursively defined towers of function fields. Such towers in turn are used to produce excellent error-correcting codes.
Prasant Singh, who will very soon defend his Ph.D. thesis in IIT Bombay, has recently been offered a H.C.Ørsted-postdoc position in the algebra group. Yesterday, he has accepted the offer. He will start his work at DTU on November 15. We are happy to welcome him into the team!
Prasant’s project deals with the investigation of Grassmann codes, codes constructed from Grassmann varieties. These highly algebraic codes exhibit a structure akin to the highly successful class of LDPC codes, which makes it possible to decode Grassmann codes using LDPC-decoding inspired methods. Last year master student Jesper Gielov Olsen demonstrated in his master project, that this decoder performs very well experimentally. Prasant will study theoretical properties of Grassmann and related codes as well as their decoding more fully.
Professor S.R. Ghorpade from IIT Bombay will visit the algebra group 4-14 June. He has visited DTU several times before and has published several articles together with the algebra group. We are looking forward to welcoming him at DTU soon!
Update to the post from January 25: Both submissions to International Symposium in Information Theory 2017 have been accepted! The papers are:
Twisted Reed-Solomon Codes, Sven Puchinger, Johan Rosenkilde, Peter Beelen. arXiv pre-print.
Decoding of Interleaved Reed-Solomon Code using Improved Power Decoding, Sven Puchinger, Johan Rosenkilde. arXiv pre-print.