Regular fat linear sets

TL;DR

This work introduces $(r,i)$-regular fat linear sets as a unifying framework encompassing scattered linear sets and $i$-clubs, and develops both general theory and explicit constructions for various $(r,i)$, including cases with $n$ composite. It analyzes two main construction paradigms that produce $(r,i)$-regular fat sets in different projective spaces, and studies the equivalence of these sets under $ ext{ΓL}$-actions. A central contribution is showing that regular fat linear sets yield three-weight rank-metric codes; using MacWilliams identities, the authors derive nontrivial bounds and illuminate weight-distribution structures. The paper also presents a detailed examination of the polynomial $\phi_{m,\sigma}$, delineating conditions under which the associated linear set is regular fat and describing the resulting weight configurations. These results advance the understanding of the geometric-coding correspondence and pose natural open problems on existence, classification, and parameter ranges for regular fat sets.

Abstract

In this work, we introduce $(r,i)$-regular fat linear sets, which are defined as linear sets containing exactly $r$ points of weight $i$ and all other points of weight one. This notion generalizes and unifies existing constructions; scattered linear sets, clubs, and other previously studied families are special cases. We present new classes of regular fat linear sets in PG$(k-1,q^n)$ for composite $n$ and study their equivalence classes. Finally, we show that regular fat linear sets naturally yield three-weight rank-metric codes, which we use to obtain bounds on their parameters.

Theorems & Definitions (58)

• Definition 2.1
• Remark 2.2
• Proposition 2.3
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9
• Corollary 2.10
• Theorem 2.11
• Theorem 3.1