SEMINAR: Groups and Combinatorics Seminar: John Sheekey, 4pm Friday 20 July in Weatherburn LT
|Groups and Combinatorics Seminar: John Sheekey, 4pm Friday 20 July in Weatherburn LT
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
Fri, 20 Jul 2018 16:00
Fri, 20 Jul 2018 17:00
Stephen Glasby <[email protected]>
Tue, 17 Jul 2018 09:46
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