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

Abstract

In 1982, Cameron and Liebler investigated certain special sets of lines in $\mathrm{PG}(3,q)$, 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. 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.