Mathematical Aspects of Information Transmission: Effective Error Correcting Codes

The project has consisted of strengthening and extending a co-operation between the departments of mathematics at IIT-Bombay and UiT-The Arctic University of Norway. The cooperation has taken place within the field of error-correcting codes. We have developed further a useful mathematical theory for how to encode and represent digital information that is sent over some channel, where the information signals are vulnerable to noise and disturbances. The goal is to make the signals more robust when exposed to these disturbances. We have contributed to finding theory and methods, such that the receivers can interpret the signals in a completely correct way, even if the signals are distorted (within limits). Typical examples of information that is encoded and sent this way are satellite communication, TV pictures, and design of registration numbers. Coding theory can also be used as a tool within cryptological settings, where the users combine these methods with other techniques, where one encrypts information. The mathematical techniques that we have used in our project are algebra, combinatorics, and algebraic geometry, which are well established and classical theories, which have proved to be very useful both in coding theory and many other areas, over the last decades. During the project period three researchers from India have been working temporarily at UiT, Tromsø as post-docs. The first one started on June 1, 2018, and left his position on May 31, 2020, in order to start in his present permanent position as assistant Professor at IIT-Hyderabad, India. The second researcher was hired as a post-doc August 1, 2019-October 31, 2021. Later that year he started in a permanent position as Assistant Professor at IIT-Jammu, India. During the period Nov. 1., 2020 - March. 31, 2022 a third young (now female) researcher from India is hired as an employee at UiT as a part of the project. Furthermore researchers from UiT and IIT-Bombay have been visiting each other for shorter periods, ranging from 10 days to 3 months. We also have had research exchenge with DTU. Lyngby, Denmark, where the two post-docs had positions before they came to Tromsø. During the project period (so far) the participants have published many articles within these themes in internationally recognized journals or lecture notes during the project period. In addition several other articles are now being refereed in similar journals for publication. Some of our articles have treated how one can find as many points as possible on various geometric objects (curves, surfaces aso.) One chooses these objects in an optimal way, such that they have a potential for construction of error-correcting codes that can represent as large amounts of information as possible, while at the same time allowing for transmission errors to be corrected. In several other articles one has treated objects and concepts from algebra and combinatorics, while in the next step utilizing them in new and innovative ways, to produce error-correcting codes.

Prosjektet har ført til forekomsten av nye og interessante metoder innenfor informasjonsteori/kodeteori, der en arbeider for å sende digitale signaler, slik at disse er robuste mot støy og kan forstås 100% korrekt av mottaker, selv om de er blitt forstyrret under sendingen. Videre har prosjektet hatt en betyydelig rekrutteringseffekt, idet unge forskere fra India har kvalifisert seg til (og fått) faste stillinger ved gode universiteter i sitt hjemland etter å ha arbeidet ved vårt prosjekt. Forskningsmiljøene ved IIT i India og UiT i Norge har videre blitt knyttet nærmere hverandre på varig basis, og vi har konkrete planer for for fortsatte fellesprosjekt i perioden framover. Langsiktige endringer vil en kunne se i form av bedret digital kommunikasjon innen en rekke samfunnsområder, og da som en del av den samlede forsknings- og utviklingsinnsats innen fagfeltet globalt. Vårt prosjekt har inngått som en naturlig og integrert del av denne innsatsen.

The project consists of creating an arena for long-lasting and stable cooperation between the two institutions within the field of coding theory. This is a field of science that is treated by the departments of informatics, or computer science, in many universities, but which is organized under the departments of mathematics at IIT Bombay, and UiT - The Arctic University of Norway (UiT-AUN). One will merge the already actively ongoing research activities within this field in a joint effort to obtain more and better results within algebraic coding theory. The results will be published as articles in international journal of high quality. This will improve the theoretical foundation for practical applications whereever error-correcting codes are used, in image processing, satellite communication, birth registers and a wide range of other areas of communication, in society at large. The results and competence developed will be a source of contact between the universities involved and enterprises/institutions in both countries, that are working with these applications. As a main part of the project two post-docs, with an understanding of methods used at the Department of Mathematics at IIT Bombay, will hold positions at UiT-AUN during parts of the period. In addition the project will consist of researchers from the two instititutions visiting each other, in part through a joint conference at IIT Bombay, also with other international participation, and in part through individual visits. The project is organized as 3 thematic work packages, where the conference in 2020 will address at least one of them, algebraic-geometric codes. The two other packages (links to algebra,combinatorics, topology, and larger classes of codes) will correspond to the activities of the two post-docs that stay in Tromsø.