The geometric representations of rank-metric codes

Official URL: https://risc01.sabanciuniv.edu/record=b2583852_(Table of contents)

In this thesis, geometric representations of rank-metric codes have been examined as well as their connection with algebraic coding theory and complexity theory. Given a vector code, we introduced an algorithm using the well-known field reduction map from projective geometry to get the corresponding rank-metric code. Following that correspondence, we revisited the codes that satisfy the analogues of the Singleton bound, called maximum rank distance(MRD) codes, and show that there is a one-to-one correspondence to finite semifields if they are additive. Given a semifield, we get a tensor associated to it. Tensor rank of various objects have been analyzed and its relation with complexity theory is explained in detail. In 1977, Kruskal proposed a lower bound on tensor rank and the codes that satisfy this bound are called minimal tensor rank(MTR) codes. We state an open problem on the existence of MTR codes deducing from the analyzed cases so far. We have solved the existence problem and proposed an attack on the characterization of all possible solutions using the algorithm Snakes and Ladders with the help of the computer algebra system GAP.