Two pointsets in $\mathrm{PG}(2,q^n)$ and the associated codes

TL;DR

This paper explores two pointsets derived from linear sets in projective geometry, analyzes their intersection patterns with lines, and studies the associated codes' weight distributions, revealing connections to their algebraic properties.

Contribution

It introduces new geometric constructions of pointsets in projective spaces and fully characterizes the weight distributions of their associated codes, linking algebraic properties to code equivalence.

Findings

The intersection patterns relate to the linear set's weight distribution.

The associated codes can have few weights when the linear set has a short weight distribution.

A connection between the linear set's Galois class and the number of inequivalent codes is established.

Abstract

In this paper we consider two pointsets in $\mathrm{PG}(2,q^n)$ arising from a linear set $L$ of rank $n$ contained in a line of $\mathrm{PG}(2,q^n)$: the first one is a linear blocking set of R\'edei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set $L$. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing $L$ to be an $\mathbb{F}_q$-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the $\Gamma\mathrm{L}$-class of $L$ and the number of inequivalent codes we can construct starting from it.