Final Report Summary - FACE (Finite and Algebraic Geometry for Error correction)
The main objective of the FACE project is to study and develop the connections between Coding Theory and non-linear geometric objects in Galois spaces, by using both theoretical and computational instruments. The study will primarily focus on particular types of non-linear geometric objects: arcs (the geometrical counterparts of error-correcting linear codes), (n,r)-arcs, caps. These objects can be viewed as projective systems or dual projective systems of points, and therefore they correspond to specific types of linear codes. These codes are deeply involved in the process of transmission of information, since they protect the information against errors occurring during the transmission process; without this protection the information received could be unusable. Linear error correcting codes are fundamental tools used to transmit digital information in many different contexts, such as military transmissions and web transmissions.
The main aim is to construct and to classify new examples and new infinite families of the geometrical counterparts of error-correcting linear codes with good parameters, i.e. correcting a large number of errors with respect to their length.
After analyzing what is known in the literature concerning Functional codes and classical polar spaces, I finished two papers about functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric and bounds on the number of rational points of algebraic hypersurfaces over finite fields. Also I submitted a paper concerning configurations of hyperplanes in finite projective spaces: this topic is related to the determination of codewords of small weight in Reed-Muller codes. Concerning the constructions of families of (n,r)-complete arcs in finite projective spaces, I studied the case of infinite families of small complete arcs in projective spaces and (n,3)-arcs in projective planes. I am still working on possible developments for infinite families of small complete (n,4)-arcs in projective planes.
Concerning the constructions of families of small complete caps in finite projective spaces, I used in particular computational methods in order to construct complete caps in projective spaces of dimension 3. Now I am starting to investigate how these caps can be described by algebraic and theoretical methods.
I started working on the construction of families of saturating sets in projective spaces and I obtained results using probabilistic methods. During this period I also had the opportunity to attend many conferences concerning different topics of my research interests, to present my results, and to meet new people with which start new collaborations. I could also continue my collaboration with some colleagues in my old University of Perugia.
Finally, part of the work has been done with the help of the COST Action IC1104 Random Network Coding and Designs over GF(q), so that my project also contributed to this COST project as proposed in my application.
Most of the results obtained during this project will be useful for the scientific community to better understand the link between Finite Geometry and Coding Theory and could be used as a starting point of other constructions or generalizations. In particular, connections with Coding Theory could be relevant in any field related to the communication of information and the protection of errors occurring during the transmission.
The main aim is to construct and to classify new examples and new infinite families of the geometrical counterparts of error-correcting linear codes with good parameters, i.e. correcting a large number of errors with respect to their length.
After analyzing what is known in the literature concerning Functional codes and classical polar spaces, I finished two papers about functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric and bounds on the number of rational points of algebraic hypersurfaces over finite fields. Also I submitted a paper concerning configurations of hyperplanes in finite projective spaces: this topic is related to the determination of codewords of small weight in Reed-Muller codes. Concerning the constructions of families of (n,r)-complete arcs in finite projective spaces, I studied the case of infinite families of small complete arcs in projective spaces and (n,3)-arcs in projective planes. I am still working on possible developments for infinite families of small complete (n,4)-arcs in projective planes.
Concerning the constructions of families of small complete caps in finite projective spaces, I used in particular computational methods in order to construct complete caps in projective spaces of dimension 3. Now I am starting to investigate how these caps can be described by algebraic and theoretical methods.
I started working on the construction of families of saturating sets in projective spaces and I obtained results using probabilistic methods. During this period I also had the opportunity to attend many conferences concerning different topics of my research interests, to present my results, and to meet new people with which start new collaborations. I could also continue my collaboration with some colleagues in my old University of Perugia.
Finally, part of the work has been done with the help of the COST Action IC1104 Random Network Coding and Designs over GF(q), so that my project also contributed to this COST project as proposed in my application.
Most of the results obtained during this project will be useful for the scientific community to better understand the link between Finite Geometry and Coding Theory and could be used as a starting point of other constructions or generalizations. In particular, connections with Coding Theory could be relevant in any field related to the communication of information and the protection of errors occurring during the transmission.