Rank-Metric Codes and Semifields

A rank-metric code is a set or subspace of matrices over a field equipped with the distance function d(X,Y)=rank(X-Y). These were first studied by Delsarte in the 1970s, who showed that a subspace of m x n matrices in which every nonzero element has rank at least d has dimension at most n(m-d+1) if m\leq n. Moreover, he showed that codes meeting this bound, known as MRD codes, exist for all parameters over any finite field.

In recent years, interest in this topic has been renewed due to new potential applications (such as random network coding, post-quantum cryptography) and connections to topics in finite geometry (such as linear sets, semifields).

In this talk we will give an overview of the constructions and applications of rank-metric codes. Particular focus will be given to the connection with finite semifields (nonassociative division algebras), and we will report on recent results regarding multiplicative complexity via calculation of the tensor rank of certain semifields.

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Link for the slides: https://cloud-ljk.imag.fr/index.php/s/crigdAR5R2kmssf