Linear sets on the projective line with complementary weights

TL;DR

This paper investigates linear sets on the projective line with two points of complementary weight, providing new examples, analyzing their equivalence, and characterizing associated linearized polynomials, thus advancing understanding of their structure and applications.

Contribution

It introduces new classes of linear sets with two points of complementary weight and studies their equivalence and defining polynomials, expanding the known examples and theoretical framework.

Findings

New examples of linear sets with two points of complementary weight

Characterization of linearized polynomials defining these linear sets

Analysis of the equivalence problem for these linear sets

Abstract

Linear sets on the projective line have attracted a lot of attention because of their link with blocking sets, KM-arcs and rank-metric codes. In this paper, we study linear sets having two points of complementary weight, that is with two points for which the sum of their weights equals the rank of the linear set. As a special case, we study those linear sets having exactly two points of weight greater than one, by showing new examples and studying their equivalence issue. Also we determine some linearized polynomials defining the linear sets recently introduced by Jena and Van de Voorde (2021).