On the stabilizer of the graph of linear functions over finite fields
Valentino Smaldore, Corrado Zanella, Ferdinando Zullo
TL;DR
The paper investigates the stabilizer $\mathbb{S}_f$ of the graph of a $\mathbb{F}_q$-linearized polynomial $f$ over $\mathbb{F}_{q^n}$ and its algebraic structure. It proves that if $f$ has low weight (intersection with every affine line is $<q^{n/2}$), then $\mathbb{S}_f$ is a field, and it characterizes many families where this holds or fails, including constructions yielding non-field stabilizers via $\operatorname{L-}q^t$- or $\operatorname{R-}q^t$-partially scattered polynomials and projection/weight-complementary cases. A central theme is the link between $\mathbb{S}_f$ and the right idealizer $R(\mathcal{C}_f)$ of the two-dimensional rank-metric code $\mathcal{C}_f=\langle x,f\rangle_{\mathbb{F}_{q^n}}$, establishing isomorphisms and transfer of field-structure results. The work further extends to MRD codes arising from partially scattered polynomials, showing their right idealizers are finite fields (size at most $q^t$) and clarifying how stabilizer properties reflect code-theoretic invariants. Overall, the results illuminate a deep connection between combinatorial geometry of graphs, linear sets, and the algebra of rank-metric codes, with practical implications for code construction and classification.
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.