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

TL;DR

This paper develops a unified theory of the degree of Boolean functions within Johnson and q-Johnson schemes, exploring algebraic and geometric properties, and establishing divisibility properties related to these functions.

Contribution

It introduces a refined definition of degree and weights of Boolean functions in association schemes, unifies various generalizations, and analyzes their algebraic and geometric properties.

Findings

Generalized the notion of degree and weights of Boolean functions in Johnson schemes

Proved a divisibility property for the sizes of Boolean functions of a given degree

Analyzed the effects of dualization and modifications on degree and weights

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.