Polymatroid theory applied to coding theory
Dr. Thomas Westerbäck
Aalto University
Recent research has proven matroid theory to be a valuable tool in several areas of coding theory when field-linear codes are considered, e.g. in distributed storage, index coding and network coding. Polymatroids, a generalization of matroids, can be used to analyse codes in general in that codes are defined as subsets of A_1 x … x A_n, where each A_i is some arbitrary finite set.
A polymatroid can be considered both as a set-combinatorial object and as a special class of polytopes associated with a submodular function. By connecting random variables to the components of a code, the code can be associated to a polymatroid via its joint entropy. Which code properties that are captured by the associated polymatroid depend on which type of code it is.
In this talk we will present how polymatroid theory can be used in order to analyse important properties of codes. Results on codes will be given in the setting of polymatroids. These results can therefore also be applied to non-code objects that are associated with polymatroids.
Since April 2014 I’m a postdoctoral researcher at the Department of Mathematics and Systems Analysis at Aalto University, Finland. In 2012 and 2013, I was a deputy lecturer for one year at KTH, Royal Institute of Technology, Sweden, and did two longer research visits to the University of New South Wales, Australia, and Aalto University. I received my M.Sc. degree in engineering and computer science and Ph.D. degree in mathematics from KTH in 2006 and 2012, respectively.
My research interests include matroid theory and generalizations thereof, algebraic and enumerative combinatorics, and applications of these theories to different areas in coding theory such as distributed storage and network coding.