On the weight distribution of linear sets with complementary weights and related constructions

TL;DR

This paper investigates the weight distribution of linear sets with complementary weights, providing criteria for point weights, a product-type construction to control weights, and methods to generate sets with diverse weights and intersection properties.

Contribution

It introduces a new product-type construction for linear sets with complementary weights, enabling precise control over their weight distribution and intersection characteristics.

Findings

Criteria for fixed weight points in linear sets

Construction of linear sets with few points of weight > 1

Linear sets with many different weights and intersection numbers

Abstract

In this paper, we continue the study of linear sets with complementary weights. We find criteria to determine the set of points of any fixed weight and use this to present particular linear sets with few points of weight more than one. We also present a product-type construction for linear sets of complementary type arising from any linear set, allowing us to control the weight distribution of the obtained linear set. Finally, we use this construction to create linear sets with many different weights, along with point sets of even type with many distinct intersection numbers.