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The degree of functions in the Johnson and q-Johnson schemes

Michael Kiermaier, Jonathan Mannaert, Alfred Wassermann

TL;DR

This work addresses generalizing Cameron-Liebler line classes beyond $PG(3,q)$ by recasting them as degree-$t$ Boolean functions in the Johnson and $q$-Johnson schemes, enabling a unified framework across ambient dimension $n$, subspace dimension $k$, and degree $t$. It introduces the degree $ ext{deg}(f)$ as the smallest $t$ with $f\in \bar{V}^{(k)}_t$ and defines $i$-weights $ ext{wt}^{(i)}_f$, tying these to the eigenstructure of the (q-)Johnson scheme via a chain of row spaces and pencils. The paper establishes a design-orthogonality correspondence linking algebraic and geometric properties, proves a divisibility property for the size of degree-$t$ functions, shows that dualization preserves degree, and analyzes the two elementary ambient-space changes that preserve or transform the degree and weight distributions. The framework recovers classical Cameron-Liebler phenomena, clarifies known degree-1 behavior, and provides a robust toolkit for analyzing $t$-antidesigns in both the Johnson and $q$-Johnson settings, with an illuminating Johnson-scheme example illustrating the theory's mechanics.

Abstract

In 1982, Cameron and Liebler investigated certain "special sets of lines" in PG(3,q), and gave several equivalent characterizations. Due to their interesting geometric and algebraic properties, these "Cameron-Liebler line classes" got much attention. Several generalizations and variants have been considered in the literature, the main directions being a variation of the dimensions of the involved spaces, and studying the analogous situation in the subset lattice. An important tool is the interpretation of the objects as Boolean functions in the "Johnson" and "q-Johnson schemes". In this article, we develop a unified theory covering all these variations. Generalized versions of algebraic and geometric properties will be investigated, having a parallel in the notion of "designs" and "antidesigns" in association schemes, which is connected to Delsarte's concept of "design-orthogonality". This leads to a natural definition of the "degree" and the "weights" of functions in the ambient scheme, refining the existing definitions. We will study the effect of dualization and of elementary modifications of the ambient space on the degree and the weights. Moreover, a divisibility property of the sizes of Boolean functions of degree t will be proven.

The degree of functions in the Johnson and q-Johnson schemes

TL;DR

This work addresses generalizing Cameron-Liebler line classes beyond by recasting them as degree- Boolean functions in the Johnson and -Johnson schemes, enabling a unified framework across ambient dimension , subspace dimension , and degree . It introduces the degree as the smallest with and defines -weights , tying these to the eigenstructure of the (q-)Johnson scheme via a chain of row spaces and pencils. The paper establishes a design-orthogonality correspondence linking algebraic and geometric properties, proves a divisibility property for the size of degree- functions, shows that dualization preserves degree, and analyzes the two elementary ambient-space changes that preserve or transform the degree and weight distributions. The framework recovers classical Cameron-Liebler phenomena, clarifies known degree-1 behavior, and provides a robust toolkit for analyzing -antidesigns in both the Johnson and -Johnson settings, with an illuminating Johnson-scheme example illustrating the theory's mechanics.

Abstract

In 1982, Cameron and Liebler investigated certain "special sets of lines" in PG(3,q), and gave several equivalent characterizations. Due to their interesting geometric and algebraic properties, these "Cameron-Liebler line classes" got much attention. Several generalizations and variants have been considered in the literature, the main directions being a variation of the dimensions of the involved spaces, and studying the analogous situation in the subset lattice. An important tool is the interpretation of the objects as Boolean functions in the "Johnson" and "q-Johnson schemes". In this article, we develop a unified theory covering all these variations. Generalized versions of algebraic and geometric properties will be investigated, having a parallel in the notion of "designs" and "antidesigns" in association schemes, which is connected to Delsarte's concept of "design-orthogonality". This leads to a natural definition of the "degree" and the "weights" of functions in the ambient scheme, refining the existing definitions. We will study the effect of dualization and of elementary modifications of the ambient space on the degree and the weights. Moreover, a divisibility property of the sizes of Boolean functions of degree t will be proven.
Paper Structure (18 sections, 32 theorems, 108 equations)

This paper contains 18 sections, 32 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Classical Cameron-Liebler line classes
  3. Generalizations
  4. Generalization of the dimension $n$ of the ambient space.
  5. Generalization of the dimension $k$ of the subspaces.
  6. The set case $q=1$.
  7. Generalization of the degree $t$.
  8. A unified generalization: Antidesigns.
  9. Structure of the paper
  10. Preliminaries
  11. $q$-numbers and $q$-binomial coefficients
  12. Subset and subspace lattices
  13. Designs
  14. Algebraic and geometric properties
  15. Weights and degree
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $a,b,u\in\{0,\ldots,n\}$ and let $B \in\genfrac{[}{]}{0pt}{}{V}{b}$. For $A\in\genfrac{[}{]}{0pt}{}{B}{a}$, we have and

Theorems & Definitions (75)

  • Lemma 2.1
  • Lemma 2.3
  • Proof 1
  • Lemma 2.6
  • Proof 2
  • Remark 3.1
  • Lemma 3.2
  • Proof 3
  • Corollary 3.3
  • Proof 4
  • ...and 65 more