An Algorithmic Approach to the Extensibility of Association Schemes

TL;DR

This paper introduces a measure called maximal height to quantify how close an association scheme is to being Schurian, and presents an algorithm to compute this height efficiently, revealing many schemes are inextensible.

Contribution

It formalizes the concept of maximal height of association schemes and develops an algorithm to determine extensibility to that height efficiently.

Findings

All non-Schurian schemes up to order 26 are inextensible.

The algorithm decides extensibility in polynomial time for fixed t.

Infinite families of inextensible schemes are constructed via tensor products.

Abstract

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,\Gamma) 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,\Gamma) 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.