Hamming Sandwiches

Abstract

AbstractWe describe primitive association schemes $${\mathfrak {X}}$$ X of degree n such that $$\textrm{Aut}({\mathfrak {X}})$$ Aut ( X ) is imprimitive and $$|\textrm{Aut}({\mathfrak {X}})| \ge \exp (n^{1/8})$$ | Aut ( X ) | ≥ 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 dth tensor power of the trivial scheme and the d-dimensional Hamming scheme. We study Hamming sandwiches in general, and exhaustively for $$d \le 8$$ d ≤ 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 $$\exp O(n^{1/8} \log n)$$ exp O ( n 1 / 8 log n ) automorphisms.