Finite semifields and rank-metric codes
Title: Finite semifields and rank-metric codes
Speaker: John Sheekey (UCD)
Date: Monday, 8th February 2016
Time: 4pm
Location: Agriculture. 1.01 (Seminar Room).
Abstract:
Finite semifields are division algebras over a finite field, where multiplication is not assumed to be associative. Non-trivial examples were first constructed by Dickson in 1906. Nowadays there are many constructions and results known, but full classification is still a long way off.
Rank-metric codes are codes where the codewords are matrices over a field, and the distance function is given by d(X,Y) = rank(X-Y). In the case where the code is in fact a subspace, and the space has optimal dimension with respect to rank-distance, we call them Maximum Rank-distance (MRD) codes. These were studied by Delsarte in the 70s and Gabidulin in the 80s, and have seen a surge in interest in recent years, mainly due to their applications in random network coding.
Semifield provide examples of MRD codes of certain parameters. In this talk we will discuss the links between semifields and codes, and show how a family of semifields due to Albert can be extended to a new family of MRD codes for all parameters. We will also discuss some open problems in both areas, mainly through the prism of linearized polynomials.
Series: Algebra
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