Classification of (v, 3)-configurations

A (v, 3)-configuration is a nondegenerate matrix of dimension v over the field 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 ≥ 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.