Constructions of optimal quaternary constant weight codes via group divisible designs

Generalized Steiner systems GS(2,k,v,g) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size g+1g+1 with minimum Hamming distance 2k−32k−3, in which each codeword has length vv and weight kk. As to the existence of a GS(2,k,v,g), a lot of work has been done for k=3k=3, while not so much is known for k=4k=4. The notion k-∗k-∗GDD was first introduced by Chen et al. and used to construct GS(2,3,v,6). The necessary condition for the existence of a 4-∗GDD(6v) is v≥14v≥14. In this paper, it is proved that there exists a 4-∗GDD(6v) for any prime power v≡3,5,7(mod8) and v≥19v≥19. By using this result, the known results on the existence of optimal quaternary constant weight codes are then extended.