Self-dual association schemes, fusions of Hamming schemes, and partial geometric designs

Partial geometric designs can be constructed from basic relations of association schemes. An infinite family of partial geometric designs were constructed from the fusion schemes of certain Hamming schemes in work by Nowak et al. (2016). A general method to create partial geometric designs from association schemes is given by Xu (2023). In this paper, we continue the research by Xu (2023). We will first study the properties and characterizations of self-dual association schemes. Then using the characterizations of self-dual association schemes and the representation theory (character tables) of commutative association schemes, we obtain characterizations and classifications of self-dual (symmetric or nonsymmetric) association schemes of rank 4 that produce as many as possible nontrivial partial geometric designs or 2-designs. In particular, for a primitive self-dual symmetric association scheme $X=\left(X,{\left\{R_i\right\}}_{0\le i\le 3}\right)$ of rank 4, if $\mid X\mid$ is a power of 3 and each of $R_1$, $R_2$, and $R_0\cup R_3$ induces a partial geometric design, then we will prove that $X$ is algebraically isomorphic to a fusion scheme of the Hamming scheme $H\left(d,3\right)$ for some odd number $d$.