Abstract We investigate unbiased weighing matrices of weight 9 and provide a construction method using mutually suitable Latin squares. For $$n \le 16$$ n ≤ 16 , we determine the maximum size among sets of mutually unbiased weighing matrices of order n and weight 9. Notably, our findings reveal that 13 is the smallest order where such pairs exist, and 16 is the first order for which a maximum class of unbiased weighing matrices is found.

A finite set X in the Euclidean unit sphere is called an s-distance set if the set of distances between any two distinct elements of X has size s. We say that t is the strength of X if X is a spherical t-design but not a spherical $(t+1)$-design. Delsarte Goethals, and Seidel gave an absolute bound for the cardinality of an s-distance set. The results of Neumaier and Cameron, Goethals, and Seidel imply that if X is a spherical 2-distance set with strength 2, then the known absolute bound for 2-distance sets can be improved. This bound is also regarded as that for a strongly regular graph with a certain condition of the Krein parameters. In this paper, we give two generalizations of this bound to spherical s-distance sets with strength t (more generally, to s-distance sets with strength t in a two-point-homogeneous space) and to Q-polynomial association schemes. First, for any s and $s-1 \leq t \leq 2s-2$, we improve the known absolute bound for the size of a spherical s-distance set with strength t. Second, for any s, we give an absolute bound for the size of a Q-polynomial association scheme of class s with some conditions of the Krein parameters.