关联方案可扩展性的数学方法
An Algorithmic Approach to the Extensibility of Association Schemes
摘要：
与高度t的预超方案相关联的关联方案被称为可扩展到高度t。Smith（1994，2007）证明了一个d阶的结合方案X=（Q，Γ）：= |Q|是Schurian的当且仅当X可扩展到高度（d-2）。在这项工作中，我们将关联方案X的最大高度t_max（X）形式化为最大正整数，使得X可扩展到高度t（我们还包括t_max（X）= \infty的可能性，其等效于t_max（X）\ge（d-2））。直观地说，最大高度提供了一个自然的度量，衡量一个关联方案与舒里亚方案的接近程度。为了计算最大高度，我们引入了关联模式扩展算法。在输入阶为d：= |Q|的关联方案X=（Q，Γ）和使得1 \le t \le（d-2）的整数t时，关联方案扩展算法在时间d^（O（t））中判定方案X是否可扩展到高度t。特别地，如果t是固定常数，则关联方案扩展算法的运行时间是X阶多项式。关联方案扩展算法被用来证明所有阶数不超过26的非Schurian关联方案都是完全不可扩展的，即它们不能扩展到一个正高度。通过结合方案的张量积，后一结果给出了完全不可扩展的结合方案的无限族的大量例子。
英文摘要：
An association scheme which is associated to a height t presuperscheme is said to be extensible to height t. Smith (1994, 2007) showed that an association scheme X=(Q,Γ) of order d:=|Q| is Schurian iff X is extensible to height (d-2). In this work, we formalize the maximal height t_max(X) of an association scheme X as the largest positive integer such that X is extensible to height t (we also include the possibility t_max(X)=\infty, which is equivalent to t_max(X)\ge (d-2)). Intuitively, the maximal height provides a natural measure of how close an association scheme is to being Schurian. For the purpose of computing the maximal height, we introduce the association scheme extension algorithm. On input an association scheme X=(Q,Γ) of order d:=|Q| and an integer t such that 1\le t\le (d-2), the association scheme extension algorithm decides in time d^(O(t)) if the scheme X is extensible to height t. In particular, if t is a fixed constant, then the running time of the association scheme extension algorithm is polynomial in the order of X. The association scheme extension algorithm is used to show that all non-Schurian association schemes up to order 26 are completely inextensible, i.e. they are not extensible to a positive height. Via the tensor product of association schemes, the latter result gives rise to a multitude of examples of infinite families of completely inextensible association schemes.
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