Degree 2 Boolean Functions on Grassmann Graphs

Abstract

We investigate the existence of Boolean degree $d$ functions on the Grassmann graph of $k$-spaces in the vector space $\mathbb{F}_q^n$. For $d=1$ several non-existence and classification results are known, and no non-trivial examples are known for $n \geq 5$. This paper focusses on providing a list of examples on the case $d=2$ in general dimension and in particular for $(n, k)=(6,3)$ and $(n,k) = (8, 4)$.We also discuss connections to the analysis of Boolean functions, regular sets/equitable bipartitions/perfect 2-colorings in graphs, $q$-analogs of designs, and permutation groups. In particular, this represents a natural generalization of Cameron-Liebler line classes.

Approximately strongly regular graphs

“…[6, §2], if Y is a regular set for all the graphs in the assocation scheme. In recent years, regular sets in graphs coming from finite geometries have received a significant amount of attention, for instance see [2,3,4,10,12,13,20,21,22,25,27,28,29,36,37,38]. Note that the investigation of regular sets in conjugacy class schemes is also of recent interest, for instance see [17,19].…”