Description of the research activity
This group contributes to the network on two different fields. On one hand, through the theory of error control codes, and on the other hand through algebraic and enumerative combinatorics.
Error control codes: Error control codes are used to detect and correct errors that may occur in data transmission or storage through eventually defective channels or storage devices that can distort the sent or stored information. For example, the atmosphere introduces errors in the transmission of images from the Meteosat satellite to Earth, different interferences in a communication by mobile phone may cause transmission errors, or reading devices need correcting algorithms for handling CDs, DVDs or USB memories. Error control codes are also used in distributed data storage in the clouds to recover lost or damaged chunks of information. The modus operandi of those codes is sending along with the original information a small amount of redundancy, so that from all the received information, one can deduce what is actually transmitted. By adding redundancy, on one side we improve the quality of the received information. But, on the other side, we augment the transmission cost.
Coding Theory aims at designing and implementing codes with good correcting capacity, while maintaining a low transmission cost, as well as designing detection and correction algorithms that allow the receiver to recover the original information. Numerical semigroups are a strong tool to parameterize the correction capacity and the redundancy of algebraic codes, as well as for enhancing decoding algorithms.
Algebraic and enumerative combinatorics: Algebraic and enumerative combinatorics are two recent disciplines. Algebraic combinatorics is the use of algebraic techniques, topology and geometry in solving combinatorial problems, or the use of combinatorial methods to address problems in these areas. Problems that can be dealt with the methods of algebraic combinatorics arise in these or other mathematical areas as well as in various parts of applied mathematics. Due to this interrelationship with different fields of mathematics, algebraic combinatorics is a confluence of a wide variety of ideas and methods.
Eumerative combinatorics aims at counting the number of elements of finite sets. An infinite collection of finite sets is considered, which is indexed by some set (usually the natural nuymbers), and the number of elements of each set is studied in general, as a function of the index. We focused on counting numerical semigroups in general and counting of numerical semigroups with certain properties, at the confluence of the two disciplines: algebraic combinatorics and enumerative combinatorics.
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