An Algorithmic Approach to the Extensibility of Association Schemes

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 tmax(X) of an association scheme X as the largest number t ∈ N such that X is extensible to height t (we also include the possibility tmax(X) = ∞, which is equivalent to tmax(X) ≥ (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 a number t ∈ N such that 1 ≤ t ≤ (d − 2), the association scheme extension algorithm decides in time d 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 any positive height t ∈ N>0. 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.