Hamming sandwiches

Abstract

We describe primitive association schemes $\mathfrak{X}$ of degree $n$ such that $\mathrm{Aut}(\mathfrak{X})$ is imprimitive and $|\mathrm{Aut}(\mathfrak{X})|\ge \exp (n^{1/8})$, 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 $d\le 8$. 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.

21 pages. Minor correction to Lemma 4.7(2). Final version incorporating referees' comments. To appear in Combinatorica

References in corpus (1)

• A characterization of Johnson and Hamming graphs and proof of Babai's conjecture