The minimum number of components in 4-regular perfect systems of difference sets

Abstract

AbstractThe number of components m in regular (m, 5, c)-systems is given in the literature to date by the inequality m ⩾ 4c − 2 (Bermond et al., “Proceedings, 18th Hungarian Combin. Colloq.”, North-Holland, Amsterdam, 1976). The case m = 4c − 2 is called extremal. It is proved that (4c − 2, 5, c)-systems do not exist. An example of a (4c, 5, c)-system with c = 2, is given. Since, in a 4-regular system, m must be even, loc. cit., it is concluded that the lower bound on the number of components is given by m >/ 4c