Authors

• Giovanni Longobardi Università degli Studi di Napoli Federico II, Italy https://orcid.org/0000-0001-9323-2043

DOI: https://doi.org/10.26493/2590-9770.1842.2f0 Keywords: Finite field, linearized polynomial, scattered polynomial, projective space, linear set, rank-metric code Abstract

The aim of this survey is to outline the state of the art in research on a class of linearized polynomials with coefficients over finite fields, known as scattered polynomials. These have been studied in several contexts, such as those described by A. Blokhuis, M. Lavrauw in 2000, and by G. Lunardon, O. Polverino in 2001. In 2016, J. Sheekey brought to light their connection to maximum rank-metric codes. This link has significantly advanced their study and investigation, sparking considerable interest in recent years.
Here, we will explore their relationship with certain subsets of the finite projective line PG(1,qⁿ) known as maximum scattered linear sets, as well as with codes made up of square matrices of order n equipped with the rank metric. We will review the known examples of scattered polynomials known to date and discuss some of their key properties. We will also address the classification of maximum scattered linear sets of the finite projective line PG(1,qⁿ) for small values of n and discuss characterization results for the examples known so far.
Finally, we will retrace how each scattered polynomial gives rise to a translation plane, as discussed by V. Casarino, G. Longobardi, C. Zanella in 2022 and by G. Longobardi, C. Zanella in 2024.

Issue Section

Open Problems on Rank Metric Codes