**Rank-Metric Codes and 3-Tensors**

**Dr. Alessandro Neri**

University of Zurich

Department of Mathematics

In coding theory, two important issues concern the encoding and the storage of a code. In the most general case, given a finite set A and a positive integer n, a block code C is a subset of A^n, endowed with a distance function (usually the Hamming one). The space of messages M is then embedded in A^n, via an injective encoding map E, such that E(M)=C. The need to have fast encoding and efficient representation of codes led to look at algebraic structures on all the defining objects (the alphabet, the space of messages and the code), and on the encoding map E. For this reason, one usually only considers the alphabet as a field of q elements, the space of messages as the vector space F_q^k, and the encoding map as a linear function into F_q^n, which leads to the study of linear codes. In this framework, we can locate the generator matrix of an [n,k]_q code C, which serves as a representation in order to store C, and also as an encoding map. However, one can also read the parameters of the defining code from its algebraic structure. This generator matrix can also be constructed for a vector rank-metric code, and in analogous ways, one can extrapolate information on the code from it.

In this talk we explain an analogous concept for rank-metric codes, which are considered as spaces of mxn matrices over a finite field F_q. This is the case of the generator tensor, which is a similar object that one can use for storing, encoding and reading the parameters of a rank-metric code. Moreover, the tensor representation leads to the investigation of a new parameter, that is the tensor rank of an [nxm, k]_q code C, that gives a measure on the storage and encoding complexity of C. This also produces an interesting relation between rank-metric codes and algebraic complexity theory.

Alessandro Neri got both his Bachelor's and Master's degree in Mathematics from University of Pisa. Since 2015, he is part of the Applied Algebra group at University of Zurich, where he received the PhD degree in July 2019, under the supervision of Prof. Joachim Rosenthal. His main research interests are Algebraic Coding Theory and Combinatorics. The main topic of his PhD project concerns rank-metric codes, their algebraic structure and their invariants.