Lara Vicino: Innovative perspectives on maximal curves and codes arising from them
The universe is conventionally described in terms of physical quantities such as energy, mass and velocity, but a quantity at least as important as these is information.
In 1948 Claude Shannon published “A mathematical theory of communication”, and this paper heralded a complete transformation in the understanding of communication. He showed that, given a noisy communication channel, reliable communication can be achieved at any rate below the channel capacity. This marked the birth of coding theory, a field of study concerned with the transmission of data across noisy channels and the recovery, or decoding, of corrupted messages. Algebra played an enormously important role in the development of this field.
Data storage, network communications, DVD-players and a lot of other devices that we use in our everyday life simply would not function without algebraic tools. Among these tools, there are Reed-Solomon (RS) codes, which are the most widely used algebraic codes. However, due to the increase in speed of transmissions and density of storage, sooner or later RS codes will have to be replaced. Algebraic Geometric (AG) codes are considered to be a possible alternative to Reed-Solomon codes, as they can correct many more error patterns and much longer bursts of errors. The main algebraic primitive used in the construction of AG codes are algebraic curves defined over a finite field.
Two important invariants of an algebraic curve C are its degree d(C) and its genus g(C). There are classical results bounding the number of points N(C) of an algebraic curve C defined over a finite field with a given genus g(C), for example the Hasse-Weil bound. Curves attaining this bound are called maximal curves. These curves are theoretically interesting as extremal objects, but also produce excellent AG codes. Goals of this project are to systematically investigate AG codes coming from maximal curves and to find other new maximal curves. Moreover, while bounds on N(C) involving g(C) are a classical subject, less is known on bounds involving d(C). A final goal of this project is therefore to study degree-maximality.