
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 of degree such that is imprimitive and , 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 th tensor power of the trivial scheme and the -dimensional Hamming scheme. We study Hamming sandwiches in general, and exhaustively for . 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…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Algebraic structures and combinatorial models
