Invariants of Rank-Metric Codes: Generalized Weights, Zeta Functions and Tensor Rank

Tensor codes were introduced by Roth in 1991 and defined to be subspaces of r-tensors where the ambient space is endowed with the tensor rank as a distance function. They are a natural generalization of the rank-metric codes introduced by Delsarte in 1978. These codes started to attract more attention in 2008 when Kötter and Kschischang proposed them as a solution to error amplification in network coding. The main theme of this dissertation is the study of combinatorial and structural properties of tensor codes. We introduce and investigate invariants of tensor codes and we classify families of them that show strong properties of rigidity and extremality. We devote the first part of this work to an overview on the body of theory developed to date for codes in the rank metric. We set up the general notation and provide the background needed in the remaining chapters. In this setting, we introduce the notion of anticodes in their general form. The approach we will use in this work will be based on these mathematical objects. In the second part of the thesis we focus on the study of algebraic invariants for vector and matrix rank-metric codes and, in particular, we generalized the theory of the zeta function for rank-metric codes developed in 2018 by Blanco-Chacón, Byrne, Duursma and Sheekey. At this point, the correct notion of optimality is needed and we classify families of codes whose invariants are either partially or entirely determined by their code parameters. As an application, we provide a generalization of the MacWilliams identities for rank-metric codes. Part of this investigation will be devoted to the study another parameter of rank-metric codes, namely their tensor rank. In 1978, Brockett and Dobkin established a connection between linear block codes and tensor rank of matrix codes, which provides a powerful tool for determining the tensor rank of codes in the rank metric. We determine the tensor rank of some space of matrices and we illustrate some consequences in coding theory. We dedicate the third part of this dissertation to invariants of tensor codes from an anticode perspective. More precisely, we initiate the theory of these algebraic objects by identifying four different classes of anticodes and investigating the related invariants. We also introduce classes of extremal tensor codes and we develop the theory of the zeta functions in the tensor case. We conclude this work on a combinatorial note by introducing the rank-metric lattices as the q-analogue of the higher-weight Dowling lattices. The latter were proposed by Dowling in 1971 in connection to a central problem in coding theory. In this part, we fully characterize the rank-metric lattices that are supersolvable and we derive closed formulas for their Whitney numbers and characteristic polynomial. Finally, we establish a connection between these lattices and the problem of distinguishing between inequivalent rank-metric codes.