We prove that a 3-GDD of type 1ⁿ k¹ ¹, where n= k, with minimum distance 3 exists for every k and such that n = k, k = 1 or 3~ (mod ~ 6), and = 1 or 3~ (mod ~ 6). These designs are of the shortest possible length (smallest number of elements) for given k and. Other constructions for such triple systems are also presented.
Tuvi Etzion (Mon,) studied this question.