论文标题
锤三明治
Hamming sandwiches
论文作者
论文摘要
我们描述了原始协会方案$ \ mathfrak {x} $ $ n $的$，以便$ \ mathrm {aut}（\ mathfrak {x}）$是不适的，并且$ | \ m m mathrm {autrm {autfrak}（\ mathfrak {x}}）| \ geq \ exp（n^{1/8}）$，与babai的猜想相矛盾。我们给出的这个示例和其他示例是Nonschurian原始相干配置（PCC）的第一个已知示例，其自动形态数量不仅仅是超数字。 我们的建筑是“锤式三明治”，协会计划夹在琐碎计划的$ d $ th张力量和$ d $维限制的锤子方案之间。我们总体上研究锤子三明治，并以$ d \ leq 8 $进行详尽的研究。 我们通过建议任何具有超过准单行数量的自动形态的PCC来修改Babai的猜想，这必须是夹在Johnson计划的张量和相应的Cameron方案之间的关联方案。如果为true，那么任何nonschurian PCC最多都具有$ \ exp o（n^{1/8} \ log n）$ automorphisms。
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.
