**Place** : LIX, room Grace Hopper

**Title** : 3-Tensors: a natural way of representing rank-metric codes

**Abstract :** 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] 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 con-

sidered as spaces of m × n 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, namely the tensor rank of an [n × m, k]

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.

It is a joint work with Eimear Byrne, Alberto Ravagnani and John Sheekey.