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

Geertrui Van de Voorde, Ferdinando Zullo

TL;DR

This work studies ${\mathbb F}_q$-linear sets in ${\mathrm PG}(1,q^n)$ with complementary weights, introducing a general criterion to determine the set of points of any fixed weight and exploiting the product-type construction $U=S\times T$ to relate weight distributions across scales. It develops explicit families (via $f\cdot\mathrm{Tr}$, Frobenius-type, and self-product forms) to control weights, and proves a key result that the weight spectrum of $L_{\mathbb{F}_{q^t}\times S_{f,\xi}}$ is determined by the weight distribution of the base set $L_f$, yielding a simple weight-enumerator relation. A recursive construction then produces linear sets with many distinct weights, including even-type sets in ${\mathrm PG}(2,q)$ with varied intersection numbers, expanding the catalog of examples beyond scattered and subline configurations. The paper closes with open problems on fully classifying weight spectra and deriving computable criteria for polynomial families, highlighting both theoretical and combinatorial gains for finite geometry and rank-metric code connections.

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.

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

TL;DR

This work studies -linear sets in with complementary weights, introducing a general criterion to determine the set of points of any fixed weight and exploiting the product-type construction to relate weight distributions across scales. It develops explicit families (via , Frobenius-type, and self-product forms) to control weights, and proves a key result that the weight spectrum of is determined by the weight distribution of the base set , yielding a simple weight-enumerator relation. A recursive construction then produces linear sets with many distinct weights, including even-type sets in with varied intersection numbers, expanding the catalog of examples beyond scattered and subline configurations. The paper closes with open problems on fully classifying weight spectra and deriving computable criteria for polynomial families, highlighting both theoretical and combinatorial gains for finite geometry and rank-metric code connections.

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.

Paper Structure

This paper contains 12 sections, 22 theorems, 64 equations.

Table of Contents

  1. Preliminaries
  2. Introduction and previous results
  3. The weight distribution of the points in a linear set with complementary weights
  4. Explicit linear sets of complementary type and their weight distributions
  5. f x Tr
  6. frob x frob
  7. f x f
  8. Linear sets determined by Fqt x f
  9. The weight distribution of linear sets determined by Fqt x f
  10. A linear set with many different weights and its associated set of even type
  11. On certain sets of even type
  12. Open problems

Key Result

Theorem 2.1

napolitano2022classifications Let $U=S\times T$, where $S$ is a $(k-r)$-dimensional ${\mathbb F}_{q}$-subspace of $\mathbb{F}_{q^n}$ with $r \leq k-r$ and $T$ is an $r$-dimensional ${\mathbb F}_q$-subspace of ${\mathbb F}_{q^n}$ and suppose that $T=\langle a_1,\ldots,a_r\rangle_{{\mathbb F}_{q}}$ fo and its size is $q^j$ with $j=\dim_{{\mathbb F}_{q}}(a_1^{-1}S\cap \ldots \cap a_r^{-1} S)$.

Theorems & Definitions (44)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Theorem 2.4
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Remark 3.4
  • ...and 34 more