Galois geometries and random network coding

Title:  Galois geometries and random network coding

Speaker:  Professor Leo Storme (Ghent University)

Date:  Monday, 7th March 2016

Time:  4pm

Location:  Agriculture. 1.01 (Seminar Room).

Abstract:

Presently, a new direction in coding theory, called Random network coding, receives a lot of attention. In random network coding, information is transmitted through a network whose topology can vary. A classical example is a wireless network where users come and go.

R. K otter and F. Kschischang proved in an inspiring article that a very good way of transmission is obtained in networks if subspace codes are used. Here, the codewords are k-dimensional vector subspaces of the n-dimensional vector space V(n; q) over the finite field of order q.

To transmit a codeword, i.e., a k-dimensional vector space, through the network, it is sufficient to transmit a basis of this k-dimensional vector space. But a k-dimensional subspace has different bases. K otter and Kschischang proved that the transmission can be optimalized if the nodes in the network transmit linear combinations of the incoming basis vectors of the k-dimensional subspace which represents the codeword.

These ideas led to many new interesting problems in coding theory and in Galois geometries. For instance, it leads to the study of sets C of k-dimensional subspaces of V(n; q), where two different k dimensional subspaces of C pairwise intersect in at most a t-dimensional subspace, for some
specified parameter t.

Since the k-dimensional subspaces of V(n; q) define (k-1)-dimensional projective subspaces of the projective space PG(n-1; q), this problem can also be investigated in a projective setting. Hence, Galois geometries can contribute to random network coding.

In this talk, we present a number of geometrical results on random network coding, thereby showing how Galois geometries can contribute to this new area in coding theory.
Analysis Seminar (TCD)


Series: Algebra