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Cameron-Liebler Line Classes with parameter $x=\frac{(q+1)^2}{3}$

Tao Feng, Koji Momihara, Morgan Rodgers, Qing Xiang, Hanlin Zou

Abstract

Cameron-Liebler line classes were introduced in \cite{CL}, and motivated by a question about orbits of collineation groups of $\PG(3,q)$. These line classes have appeared in different contexts under disguised names such as Boolean degree one functions, regular codes of covering radius one, and tight sets. In this paper we construct an infinite family of Cameron-Liebler line classes in $\PG(3,q)$ with new parameter $x=(q+1)^2/3$ for all prime powers $q$ congruent to 2 modulo 3. The examples obtained when $q$ is an odd power of two represent the first infinite family of Cameron-Liebler line classes in $\PG(3,q)$, $q$ even.

Cameron-Liebler Line Classes with parameter $x=\frac{(q+1)^2}{3}$

Abstract

Cameron-Liebler line classes were introduced in \cite{CL}, and motivated by a question about orbits of collineation groups of . These line classes have appeared in different contexts under disguised names such as Boolean degree one functions, regular codes of covering radius one, and tight sets. In this paper we construct an infinite family of Cameron-Liebler line classes in with new parameter for all prime powers congruent to 2 modulo 3. The examples obtained when is an odd power of two represent the first infinite family of Cameron-Liebler line classes in , even.

Paper Structure

This paper contains 11 sections, 24 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cameron-Liebler line classes in $\mathrm{PG}(3,q)$ and tight sets in $\mathcal{Q}^+(5,q)$
  4. Gauss sums
  5. Cubic polynomials over $\mathbb{F}_q$
  6. Cameron-Liebler line classes with parameter $x=(q+1)^2/3$
  7. The set $E$
  8. The set $\mathcal{M}$
  9. Proof of the main theorem
  10. The stabilizer of ${\mathcal{M}}$ in $\textup{P}\Omega(V)$
  11. Concluding Remarks

Key Result

Lemma \oldthetheorem

By orthogonality of characters, the canonical additive character $\psi_{\mathbb{F}_q}$ of $\mathbb{F}_q$ can be expressed as a linear combination of the multiplicative characters: where $\widehat{\mathbb{F}_q^*}$ is the character group of $\mathbb{F}_q^\ast$. On the other hand, each nontrivial multiplicative character $\chi$ of $\mathbb{F}_q$ can also be expressed as a linear combination of the a

Theorems & Definitions (45)

  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem: EHKX1999
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 35 more