Constructions of optimal quaternary constant weight codes via group divisible designs

Abstract

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 + 1 with minimum Hamming distance 2 k − 3 , in which each codeword has length v and weight k . As to the existence of a GS ( 2 , k , v , g ) , a lot of work has been done for k = 3 , while not so much is known for k = 4 . The notion 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 ( 6 v ) is v ≥ 14 . In this paper, it is proved that there exists a 4 - ∗ GDD ( 6 v ) for any prime power v ≡ 3 , 5 , 7 ( mod 8 ) and v ≥ 19 . By using this result, the known results on the existence of optimal quaternary constant weight codes are then extended.