[Paper Review] The weight distributions of linear sets in $\mathrm{PG}(1,q^5)$

Maarten De Boeck, Geertrui Van de Voorde|arXiv (Cornell University)|Jun 8, 2020

Finite Group Theory Research5 references4 citations

TL;DR

This paper determines all possible weight distributions of $ F_q$-linear sets in $ PG(1,q^5)$, proving that no $ F_q$-linear sets of rank 5 can be $2$-clubs and that no such sets contain few points of weight 2. The results fully classify the possible weight distributions for linear sets of rank 5 in this projective space.

ABSTRACT

In this paper, we determine all possible weight distributions of $\mathbb{F}_q$-linear sets properly contained in $\mathrm{PG}(1,q^5)$. In particular, we show that there exist no $2$-clubs of rank $5$, and more generally, that there are no $\mathbb{F}_q$-linear sets of rank $5$ containing few points of weight $2$.

Motivation & Objective

• To fully determine all possible weight distributions of $ F_q$-linear sets of rank 5 in $ PG(1,q^5)$.
• To investigate the existence of $2$-clubs—linear sets with only one point of weight 1 and the rest of weight 2—within $ PG(1,q^5)$.
• To examine whether $ F_q$-linear sets of rank 5 can contain only a small number of points of weight 2.
• To provide a complete classification of weight distributions for $ F_q$-linear sets in this specific projective space.

Proposed method

• Utilizing the structure of $ F_q$-linear sets in projective geometry, particularly in $ PG(1,q^5)$, to analyze their weight distributions.
• Applying combinatorial and algebraic techniques to constrain the number and distribution of points of each weight in linear sets of rank 5.
• Analyzing the intersection properties of linear sets with lines in $ PG(1,q^5)$ to derive constraints on weight configurations.
• Employing field reduction and duality techniques to relate linear sets to their associated subspaces over $ F_q$.
• Using known results on the maximum number of points of a given weight in linear sets to rule out specific configurations.
• Proving non-existence results via contradiction and structural analysis of the underlying vector subspaces.

Experimental results

Research questions

• RQ1 What are all possible weight distributions of $ F_q$-linear sets of rank 5 in $ PG(1,q^5)$?
• RQ2 Can there exist a $2$-club of rank 5 in $ PG(1,q^5)$, i.e., a linear set with exactly one point of weight 1 and all others of weight 2?
• RQ3 Do there exist $ F_q$-linear sets of rank 5 in $ PG(1,q^5)$ that contain only a small number of points of weight 2?
• RQ4 What structural constraints prevent certain weight distributions from occurring in rank 5 linear sets?

Key findings

• There are no $2$-clubs of rank 5 in $ PG(1,q^5)$, meaning no such linear set can have exactly one point of weight 1 and all others of weight 2.
• No $ F_q$-linear sets of rank 5 in $ PG(1,q^5)$ can contain only a few points of weight 2; such configurations are ruled out by structural constraints.
• The complete set of possible weight distributions for $ F_q$-linear sets of rank 5 in $ PG(1,q^5)$ is fully determined and classified.
• The analysis shows that the geometry and field structure of $ PG(1,q^5)$ impose strong restrictions on the possible weight configurations of linear sets of rank 5.