Hamming sandwiches

TL;DR

This paper constructs new primitive association schemes called Hamming sandwiches that challenge existing conjectures by exhibiting nonschurian primitive coherent configurations with super-polynomial automorphism groups.

Contribution

It introduces the concept of Hamming sandwiches, providing the first known examples of nonschurian primitive coherent configurations with super-quasipolynomial automorphism groups, and revises Babai's conjecture.

Findings

Hamming sandwiches can have automorphism groups of size at least exp(n^{1/8}).

First examples of nonschurian primitive coherent configurations with more than quasipolynomial automorphisms.

Revised conjecture limits nonschurian PCC automorphisms to exp(O(n^{1/8} log n)).

Abstract

We describe primitive association schemes $\mathfrak{X}$ of degree $n$ such that $\mathrm{Aut}(\mathfrak{X})$ is imprimitive and $|\mathrm{Aut}(\mathfrak{X})| \geq \exp(n^{1/8})$, contradicting a conjecture of Babai. This and other examples we give are the first known examples of nonschurian primitive coherent configurations (PCC) with more than a quasipolynomial number of automorphisms. Our constructions are "Hamming sandwiches", association schemes sandwiched between the $d$th tensor power of the trivial scheme and the $d$-dimensional Hamming scheme. We study Hamming sandwiches in general, and exhaustively for $d \leq 8$. We revise Babai's conjecture by suggesting that any PCC with more than a quasipolynomial number of automorphisms must be an association scheme sandwiched between a tensor power of a Johnson scheme and the corresponding full Cameron scheme. If true, it follows that any nonschurian PCC has at most $\exp O(n^{1/8} \log n)$ automorphisms.