Near-factorizations in association schemes

Source abstract

We formally initiate the study of $λ$-fold $(s,t)$ near-factorizations in association schemes. Namely, given an association $(X, \mathcal{A})$, we consider the existence of a factorization of $λ(J-I)$ into 01-matrices $S$ and $T$ with the constraint that $S$ and $T$ must belong to the adjacency algebra of $(X, \mathcal{A})$. We establish basic properties of $λ$-fold $(s,t)$ near-factorizations in association schemes and calculate bounds for $λ$, $s$, and $t$ relative to the order of the association scheme. We completely determine all near-factorizations in symmetric and asymmetric 2-class association schemes. Furthermore, we construct near-factorizations in certain Hamming schemes, cyclotomic association schemes, Schurian schemes on small primitive groups, and the folded cube and halved cube association schemes. Finally, we establish that certain Hamming schemes, Johnson schemes, and Grassmann schemes do not admit a $λ$-fold near-factorization.