On the stabilizer of the graph of linear functions over finite fields

Abstract

In this paper we will study the action of ${\mathbb{F}}_{q^n}^{2\times 2}$ on the graph of an ${\mathbb{F}}_q$-linear function of ${\mathbb{F}}_{q^n}$ into itself. In particular we will see that, under certain combinatorial assumptions, its stabilizer (together with the sum and product of matrices) is a field. We will also see some examples for which this does not happen. Moreover, we will establish a connection between such a stabilizer and the right idealizer of the rank-metric code defined by the linear function and give some structural results in the case in which the polynomials are partially scattered.