Visit of associate professor Alp Bassa

Associate professor Alp Bassa from Boğaziçi University in Istanbul (Turkey) is visiting the algebra group at DTU in the period 17-25 august. He will work with Peter Beelen and Nurdagül Anbar on various problems in function fields.

Sudhir R. Ghorpade from the Department of Mathematics at IIT-Bombay is visiting the algebra group this week. He will together with Peter Beelen and Mrinmoy Datta finish a paper in which several generalized Hamming weights of projective Reed-Muller codes are determined. Compared to the current state-of-the-art in this area, the paper will mark significant progress.

Sudhir has visited DTU several times and has coauthored several papers with Peter and other authors from DTU before. You can see us working on our current project in the picture above.

Peter Beelen will give a talk on Monday June 6 at the 2016 SIAM conference on discrete mathematics in Atlanta, USA.

He will talk about his recent progress, obtained together with postdoc Mrinmoy Datta and Prof. Sudhir Ghorpade from IIT-Bombay, on the determination of the generalized Hamming weights of projective Reed-Muller codes. Knowing these weights helps in finding the answer to a very fundamental question in algebra: “How many solutions can a system of polynomial equations have over a finite field?”

Postdoctoral researcher Nurdagül Anbar will give a talk on June 3rd on the N-cube days at KU. At this workshop several researchers on number theory will present their work. Nurdagül will talk about recent results on the asymptotic p-rank of towers of function fields.

Starting the 1st of June, Mrinmoy Datta will become a member of the algebra group in the section for mathematics at DTU Compute. He obtained a prestigious two-year postdoctoral research grant from DFF-FNU for the project Intersection of hypersurfaces: the fundament of algebraic coding theory (Grant No. 6108-00362).

Mrinmoy obtained his Ph.D. degree in mathematics at IIT-Bombay (India), under the supervision of Prof. S.R. Ghorpade. The algebra group looks forward to working together with Mrinmoy in the coming two years!

The article “The structure of dual Grassmann codes” by Peter Beelen and Fernando Piñero has just appeared in the high ranking journal Designs, Codes and Cryptography (DCC). Co-author Fernando Piñero is an old acquaintance of the algebra group: Fernando did his Ph.D. between 2011-2014 supervised by Peter Beelen and Prof. Emeritus Tom Høholdt. After his Ph.D. Fernando became a postdoc at the IIT-Bombay, India. Currently he is working at the UPR (University Puerto Rico).

Keywords: factorization, prime numbers, number theory

Prerequisites: 01018: Discrete Mathematics 2

The security of the RSA cryptosystem depends on the hardness of the integer factoring problem. In general this is a hard problem for which no fast algorithms are known. More precisely, given a natural number \(N \le 10^n\) there currently does not exist an algorithm that would for any such \(N\) finds all its prime factors in time polynomial in n.

While naïve algorithms such as finding prime factors of n by trial and error, have exponential running time in \(n\), there also exist factoring algorithms that have so-called subexponential running time in \(n\). One example of such an integer factoring algorithm is the quadratic sieve.

The currently fastest integer factoring algorithm is the famous number field sieve. The name of this algorithm comes from number fields: finite dimensional algebraic extension fields of \(\mathbb{Q}\). In this project you will learn about number fields and several rings contained in them as well as their relation to the number field sieve.

Keywords: coding theory, root finding, decoding, fast algorithms

Prerequisites: 01405: Algebraic Coding Theory

In hardware implementations of BCH decoders, error location is bottleneck, i.e. finding the error-locator polynomial and afterwards, the roots of the error locator polynomial. In high-density optical transmission, one would like to be able to correct up to 5 errors as fast as possible. In this project we will consider the following question:

How can we perform decoding for a 5-error-correcting BCH code as fast as possible?

A strategy to answer this question is the following: To speed up finding the error-locator polynomial, one can precompute a so-called generic error-locator polynomial, i.e. one can find an expression containing the syndromes as additional variables, such that for each actual value of the syndromes one obtains the correct error-locator polynomial from the generic error-locator polynomial by substituting the syndrome values.

Such expressions are too lengthy in general, but if the number of errors one needs to correct is modest this is more feasible.. To find the roots of the polynomial of degree 5, the strategy is to use suitably chosen transformations to bring any given polynomial in a standard form. Tricks similar (but algebraically more interesting) to completing the square are generalized and used for this purpose. Afterwards a strategy will be developed to solve the remaining standard cases, among others using table lookup for special cases.

During 4-11 March, Peter Beelen visited Boğaziçi university in Istanbul to work with his long-time collaborator Alp Bassa. Together with post doctoral researcher Nurdagül Anbar (currently working at DTU), they finished a paper in which several existing towers of function fields are explained using the theory of Drinfeld modules.

Peter also gave a talk on The order bound for AG codes at the Istanbul center for mathematical sciences (IMBM) located on the Boğaziçi university campus.