SEMINAR:
Groups and Combinatorics Seminar: John Sheekey, 4pm Friday 20 July in Weatherburn LT
Fri, 20 Jul 2018 16:00
Speaker: John Sheekey (University College Dublin)
Title: Subspaces of Matrices over Finite Fields with Restricted Rank
Time and place: 4pm Friday 20 Jul 2018, Weatherburn LT
Abstract: Rank-metric codes are codes consisting of matrices over a
finite field, with the distance between two matrices defined as the rank
of their difference. Delsarte showed that if U is a subspace of m x n
matrices in which the rank of every nonzero element is at least k, then
the dimension of U is at most n(m-k+1). Linear maximum rank-distance
(MRD) codes are subspaces obtaining this bound. They are the rank-metric
analogue of MDS codes. Delsarte showed the existence of examples for all
parameters. Interest in the topic has increased in recent years due to
potential applications, for example in random network coding, and in
cryptography.
Finite semifields are division algebras over a finite field, where
multiplication is not assumed to be associative. Some constructions are
known, but classification remains a difficult open problem. They have
been studied in part due to their connections to objects in finite
geometry such as projective planes, spreads, ovoids, and flocks.
Semifields also correspond to n-dimensional subspaces of n x n matrices
where every nonzero element is invertible; i.e. MRD codes with minimum
distance n.
In this talk we will give an overview of the known constructions and
classifications of semifields and MRD codes. We will present recent
algebraic constructions using linearized polynomials and skew-polynomial
rings, which constitutes the largest known family.
Past and future seminars may be found at https://www.maths.uwa.edu.au/~gl
asby/GroupsAndCombinatoricsSeminar/S18.html
For more information:
Stephen Glasby
glasbys@gmail.com
Starts : Fri, 20 Jul 2018 16:00
Ends : Fri, 20 Jul 2018 17:00
Last Updated : Tue, 17 Jul 2018 09:46