Classification of $(v,3)$-Configurations

A $(v,3)$-configuration is a nondegenerate matrix of dimension $v$ over the field $\mathrm{GF}(2)$ considered up to permutation of rows and columns and containing exactly three $1$'s in the rows and columns, while the inverse matrix has also exactly three $1$'s in the rows and columns. It is proved that, for each even $v\ge 4$, there is only one indecomposable $(v,3)$-configuration, while, for odd $v$, there are no such configurations, the only exception being the unique $(5,3)$-configuration.