On the existence of (v, 4, 3, 1)-BHDs

We prove that it is necessary and sufficient for a (v, 4, 3, 1)-BHD (balanced generalized handcuffed design) to exist that either v = 6n + 2 (n integer ⩾ 1) or v = 6n + 4 (n integer ⩾ 0).