It is a well-known property of the BIB design that all treatment effects are estimated with the same accuracy, i.e., that the variances of the estimates of the treatment effects are all equal and their covariances are also all equal. We show that the converse is also true. If the estimates of the treatment effects in an incomplete block design all have the same variances and the same covariances, then the design is a BIB.
"A Note on the Balanced Incomplete Block Designs." Ann. Math. Statist. 27 (3) 842 - 846, September, 1956. https://doi.org/10.1214/aoms/1177728192